COMPANY SUPPORT AND IRREVOCABLE PROXY AGREEMENT

Exhibit 10.6

COMPANY SUPPORT AND IRREVOCABLE PROXY AGREEMENT

SUPPORT AGREEMENT (this “Agreement”) dated as of April 29, 2013, by and among Contango Oil & Gas Company, a Delaware corporation (“Parent”), Contango Acquisition, Inc., a Delaware corporation and wholly-owned subsidiary of Parent (“Merger Sub”), and Jay S. Mengle (“Stockholder”).

WHEREAS, Stockholder is a stockholder of Crimson Exploration Inc., a Delaware corporation (the “Company”);

WHEREAS, as of the date hereof, Stockholder is the beneficial owner (as defined in Rule 13d-3 under the Exchange Act) of 340,581 shares of common stock, par value $0.001 per share, of the Company (“Company Common Stock”) (the “Original Shares” and, together with any additional shares of Company Common Stock pursuant to Section 1.5 hereof, the “Subject Shares”);

WHEREAS, contemporaneously with the execution and delivery of this Agreement, Parent, Merger Sub and the Company have entered into an Agreement and Plan of Merger (the “Merger Agreement”), providing for, among other things, the merger of Merger Sub with and into the Company, with the Company continuing as the surviving corporation and as a wholly owned subsidiary of Parent (the “Merger”);

WHEREAS, in order to induce Parent and Merger Sub to enter into the Merger Agreement, Stockholder has agreed to enter into this Agreement; and

WHEREAS, capitalized terms used but not otherwise defined herein shall have the respective meanings ascribed to such terms in the Merger Agreement.

NOW, THEREFORE, in consideration of the premises and for other good and valuable consideration given to each party hereto, the receipt of which is hereby acknowledged, the parties hereto agree as follows:

ARTICLE I

AGREEMENT TO VOTE

Section 1.1 Voting of Subject Shares; Irrevocable Proxy.

(a) Stockholder agrees to vote (or cause the holder of record on any applicable record date to vote) all of the Subject Shares in connection with any meeting of the stockholders of the Company (including any adjournment or postponement thereof) or any action by written consent in lieu of a meeting of stockholders of the Company (i) in favor of the approval of the Merger Agreement, the approval of the Merger and the other transactions contemplated by the Merger Agreement and the approval of any other matter that is required to be approved by the stockholders of the Company in order to effect the transactions contemplated by the Merger Agreement (including any proposal to adjourn or postpone a meeting of the stockholders of the Company to a later date if there are not sufficient votes to approve the

Merger Agreement); and (ii) against any other action that (A) is an agreement or arrangement constituting or related to any Acquisition Proposal or Alternative Transaction, (B) would result in a liquidation, dissolution, recapitalization, extraordinary dividend or other significant corporate reorganization of the Company; or (C) would reasonably be expected to interfere with or delay the consummation of the Merger and the other transactions contemplated by the Merger Agreement, and in connection therewith, Stockholder shall execute any documents which are necessary or appropriate in order to effectuate the foregoing. Stockholder shall (or shall cause the holder of record on any applicable record date to) be present (in person or by proxy) at any meeting of stockholders of the Company (including any adjournment or postponement thereof) called to approve the Merger Agreement or otherwise cause the Subject Shares to be counted as present thereat for purposes of establishing a quorum.

(b) In furtherance of the foregoing, Stockholder hereby irrevocably grants to, and appoints, until the termination of this Agreement in accordance with Section 2.3, Parent and any person or persons designated in writing by Parent, and each of them individually, as Stockholder’s proxy and attorney-in-fact (with full power of substitution), for and in the name, place and stead of Stockholder, to vote or grant a written consent in respect of all of the Subject Shares, or execute and deliver a proxy to vote or grant a written consent in respect of the Subject Shares, on the matters and in the manner specified in Section 1.1(a) of this Agreement. Stockholder hereby affirms that such irrevocable proxy is given in connection with, and in consideration of, the execution of the Merger Agreement by Parent and Merger Sub, and that such irrevocable proxy is given to secure the performance of the duties of Stockholder under this Agreement. Stockholder hereby further affirms that such proxy is irrevocable and is coupled with an interest sufficient in law to support an irrevocable power and may under no circumstances be revoked. Such proxy is executed and intended to be irrevocable in accordance with the provisions of Section 212 of the DGCL until the termination of this Agreement in accordance with Section 2.3.

Section 1.2 No Transfers; No Inconsistent Arrangements. Except as provided hereunder or under the Merger Agreement, Stockholder shall not, directly or indirectly, (i) transfer (which term shall include any sale, assignment, gift, pledge, hypothecation or other disposition), or consent to or permit any such transfer of, any or all of the Subject Shares or any interest therein (except where the transferee or third party agrees in writing to be bound by the terms hereof), or create or permit to exist any Lien that would prevent Stockholder from voting the Subject Shares in accordance with this Agreement or from complying with its other obligations under this Agreement, other than any restrictions imposed by applicable Law on any such Subject Shares; (ii) enter into any contracts inconsistent with the terms of this Agreement with respect to any transfer of Subject Shares or any interest therein,; (iii) grant or permit the grant of any proxy, power of attorney or other authorization in or with respect to the Subject Shares relating to the subject matter hereof; (iv) deposit or permit the deposit of the Subject Shares into a voting trust or enter into a voting agreement or arrangement with respect to the Subject Shares; or (v) take or permit any other action that would in any way restrict, limit or interfere with the performance of its obligations hereunder or the transactions contemplated hereby (any of the actions set forth in clauses (i) through (v) above, and any conversion, exchange or other disposition of the Subject Shares in an Alternative Transaction being referred to in this Agreement as a “Transfer”). To the extent the Subject Shares are represented by certificates, Stockholder shall make available to the Company such certificates in order for the Company to mark such certificates with legends required by the DGCL regarding the foregoing Transfer restrictions.

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Section 1.3 Non-Solicitation. Stockholder shall not, and shall direct and use commercially reasonable efforts to cause its Affiliates and Representatives not to, directly or indirectly, initiate, solicit or knowingly encourage (including by way of furnishing confidential information) any Third Party to make an Acquisition Proposal or assist any Third Party in preparing or soliciting an offer relating in any way to an Acquisition Proposal; provided, however, that consistent with the provisions of Section 2.14 below, nothing herein shall limit or prohibit Stockholder or any of its Representatives, in his or her capacity as an officer or director of the Company, from taking any action or failing to take any action in such capacity.

Section 1.4 Documentation and Information. Stockholder (i) consents to and authorizes the publication and disclosure by Parent, Merger Sub or the Company of Stockholder’s identity and holding of Subject Shares, and the nature of its commitments, arrangements and understandings under this Agreement, in any press release, the Joint Proxy Statement, the Form S-4 and any other disclosure document required in connection with the Merger Agreement, the Merger and any transactions contemplated by the Merger Agreement, and (ii) agrees to give to Parent as promptly as practicable any information related to the foregoing that Parent may reasonably require for the preparation of any such disclosure documents. Stockholder agrees to notify Parent as promptly as practicable of any required corrections with respect to any written information supplied by Stockholder specifically for use in any such disclosure document, if and to the extent Stockholder becomes aware that any such information shall have become false or misleading in any material respect.

Section 1.5 Changes to Subject Shares. Stockholder agrees that all shares of Company Common Stock that Stockholder purchases, acquires the right to vote or otherwise acquires beneficial ownership (as defined in Rule 13d-3 under the Exchange Act) of after the execution of this Agreement shall be subject to the terms of this Agreement and shall constitute “Subject Shares” for all purposes of this Agreement. In the event of any stock dividend or distribution, or any change to the Subject Shares by reason of any stock dividend or distribution, split-up, recapitalization, combination, exchange of shares or any other similar transaction, the term “Subject Shares” as used in this Agreement shall be deemed to refer to and include the Subject Shares and all such stock dividends and distributions and any securities into which or for which any or all of the Subject Shares may be changed or exchanged or which are received in the relevant transaction.

Section 1.6 Representations and Warranties. Stockholder represents and warrants to Parent and Merger Sub as follows:

(a) Stockholder (i) is the sole beneficial owner (within the meaning of Rule 13d-3 under the Exchange Act) of, and has, and at the time of the Company Stockholder Meeting will have, good and marketable title to the Subject Shares, free and clear of any and all Liens, proxies, voting trusts or agreements, options, rights, understandings or arrangements or any other encumbrances whatsoever on title, transfer, or exercise of any rights of a stockholder in respect of such Subject Shares (collectively, “Encumbrances”) except for Encumbrances arising (A) hereunder or (B) from the qualification of any Subject Shares as Company Restricted

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Stock; (ii) does not own, of record or beneficially, any shares of capital stock of the Company (or rights to acquire any such shares) other than the Subject Shares; and (iii) has the sole right to vote and dispose of, and holds sole power to issue instructions with respect to, the matters set forth in this Agreement with no material limitations, qualifications or other restrictions on such rights, subject to applicable federal securities law and the terms of this Agreement and except for any such restrictions arising from the qualification of any Subject Shares as Company Restricted Stock.

(b) This Agreement has been duly and validly executed and delivered by Stockholder and, assuming this Agreement constitutes a valid and binding obligation of each of Parent and Merger Sub, constitutes a legal, valid and binding agreement of Stockholder enforceable against Stockholder in accordance with its terms, subject to the Bankruptcy and Equity Exception.

(c) The execution, delivery and performance by Stockholder of this Agreement and the consummation of the transactions contemplated hereby do not and will not (i) conflict with, or result in the breach or termination of or constitute a default (with or without the giving of notice or the lapse of time or both) under (A) to the extent applicable, any provisions of the organizational documents of Stockholder; (B) any note, bond, mortgage, indenture, contract, agreement, lease, license, permit or other instrument or obligation of any kind to which Stockholder is a party or by which the Subject Shares are bound; or (ii) violate, or require any consent, approval, or notice under any provision of any judgment, order or decree or any federal, state, local or foreign statute or Law applicable to Stockholder or any of the Subject Shares.

ARTICLE II

MISCELLANEOUS

Section 2.1 Notices. Any notices or other communications required or permitted under, or otherwise given in connection with, this Agreement shall be in writing and shall be deemed to have been duly given (a) when delivered or sent if delivered in person or sent by facsimile transmission (provided confirmation of facsimile transmission is obtained); (b) on the fifth Business Day after dispatch by registered or certified mail (return receipt requested and first-class postage prepaid); or (c) on the next Business Day if transmitted by national overnight courier (with proof of service), in each case as follows: (i) if to Stockholder, to the address set forth below Stockholder’s name on the signature page hereto, and (ii) if to Parent or Merger Sub, in accordance with Section 9.2 of the Merger Agreement, or to such other Persons, addresses or facsimile numbers as may be designated in writing to each other party hereto by the Person entitled to receive such communication as provided above.

Section 2.2 Further Assurances. Stockholder will, from time to time, execute and deliver, or cause to be executed and delivered, such additional documents as Parent or Merger Sub may reasonably request for the purpose of effectively carrying out the transactions contemplated by this Agreement.

Section 2.3 Termination. This Agreement shall terminate in its entirety upon the earliest to occur of (i) the termination of the Merger Agreement in accordance with its terms,

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(ii) the Effective Time; and (iii) any reduction of the Merger Consideration or change in the form of the Merger Consideration; provided, however, that the provisions of this Article II (Miscellaneous) shall survive any termination of this Agreement. In the event of termination of this Agreement, this Agreement shall become void and of no effect with no liability on the part of any party hereto; provided, however, that the termination of this Agreement shall not prevent any party hereto from seeking any remedies (at law or in equity) against any other party hereto for such party’s breach of any of the terms of this Agreement occurring prior to such termination.

Section 2.4 Amendments and Waivers.

(a) This Agreement may not be amended except by an instrument in writing signed on behalf of each of the parties hereto.

(b) Any failure of any of the parties hereto to comply with any obligation, covenant, agreement or condition herein may be waived by the party or parties hereto entitled to the benefits thereof only by a written instrument signed by the party expressly granting such waiver, which expressly states that it is intended to waive a right hereunder, but such waiver or failure to insist upon strict compliance with such obligation, covenant, agreement or condition shall not operate as a waiver of, or estoppel with respect to, any subsequent or other failure.

Section 2.5 Expenses. All costs and expenses incurred in connection with this Agreement shall be paid by the party incurring such costs and expenses, whether or not the transactions contemplated by this Agreement or the Merger Agreement are consummated.

Section 2.6 Binding Effect; Benefit; Assignment. Neither this Agreement nor any of the rights, interests or obligations hereunder shall be assigned by any of the parties hereto, in whole or in part (whether by operation of Law or otherwise), without the prior written consent of the other party, and any attempt to make any such assignment without such consent shall be null and void. Subject to the preceding sentence, this Agreement will be binding upon, inure to the benefit of and be enforceable by the parties hereto and their respective successors and assigns.

Section 2.7 Governing Law. This Agreement shall be governed and construed in accordance with the laws of the State of Delaware (without giving effect to choice of law principles thereof).

Section 2.8 Counterparts. This Agreement may be executed in one or more counterparts, all of which shall be considered one and the same agreement and shall become effective when one or more counterparts have been signed by each of the parties hereto and delivered to the other party, it being understood that each party need not sign the same counterpart. Signatures to this Agreement transmitted by facsimile transmission, by electronic mail in “portable document format” (“.pdf”) form, or by any other electronic means intended to preserve the original graphic and pictorial appearance of a document, shall have the same effect as physical delivery of the paper document bearing the original signature.

Section 2.9 Jurisdiction. Each of the parties hereto irrevocably agrees that any legal action or proceeding with respect to this Agreement or for recognition and enforcement of any judgment in respect hereof brought by any other party or its successors or assigns may be

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brought and determined exclusively in the Court of Chancery of the State of Delaware or, if under applicable Law exclusive jurisdiction over such matter is vested in the federal courts, any court of the United States located in the State of Delaware, and each of the parties hereto hereby irrevocably submits with regard to any such action or proceeding for itself and in respect to its property, generally and unconditionally, to the exclusive jurisdiction of the aforesaid courts and agrees that it will not bring any legal action or proceeding with respect to this Agreement or for recognition and enforcement of any judgment in respect hereof in any court other than the aforesaid courts. Each of the parties hereto hereby irrevocably waives, and agrees not to assert, by way of motion, as a defense, counterclaim or otherwise, in any action or proceeding with respect to this Agreement or for recognition and enforcement of any judgment in respect hereof, (a) any claim that it is not personally subject to the jurisdiction of the above-named courts for any reason other than the failure to lawfully serve process, (b) that it or its property is exempt or immune from jurisdiction of any such court or from any legal process commenced in such courts (whether through service of notice, attachment prior to judgment, attachment in aid of execution of judgment, execution of judgment or otherwise), and (c) to the fullest extent permitted by applicable Law, that (i) the Action in any such court is brought in an inconvenient forum, (ii) the venue of such suit, action or proceeding is improper and (iii) this Agreement, or the subject matter hereof, may not be enforced in or by such courts.

Section 2.10 Service of Process. Each party hereto irrevocably consents to service of process in the manner provided for notices in Section 2.1; provided, however, that nothing in this Agreement shall affect the right of any party hereto to serve process in any other manner permitted by Law.

Section 2.11 Entire Agreement; Third Party Beneficiaries. This Agreement (a) constitutes the entire agreement and supersedes all prior agreements and understandings, both written and oral, among the parties hereto with respect to the subject matter hereof; and (b) shall be binding upon and inure solely to the benefit of each party hereto, and nothing in this Agreement, express or implied, is intended to or shall confer upon any other Person any right, benefit or remedy of any nature whatsoever under or by reason of this Agreement.

Section 2.12 Severability. If any term, provision, covenant or restriction of this Agreement is held by a court of competent jurisdiction or other Governmental Entity to be invalid, void, unenforceable or against its regulatory policy, the remainder of the terms, provisions, covenants and restrictions of this Agreement shall remain in full force and effect and shall in no way be affected, impaired or invalidated.

Section 2.13 Specific Performance. The parties hereto agree that irreparable damage would occur if any provision of this Agreement were not performed in accordance with the terms hereof or otherwise breached and that monetary damages, even if available, would not be an adequate remedy therefor and therefore fully intend for specific performance to be the principal remedy for breaches of this Agreement. It is accordingly agreed that, prior to termination of this Agreement pursuant to Section 2.3, the parties hereto shall be entitled to an injunction or injunctions to prevent breaches of this Agreement or to enforce specifically the performance of the terms and provisions hereof. The parties hereto further agree not to assert that a remedy of specific enforcement is unenforceable, invalid, contrary to Law or inequitable for any reason, nor to object to a remedy of specific performance on the basis that a remedy of

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monetary damages would provide an adequate remedy for any such breach. The parties hereto further acknowledge and agree that the agreements contained in this Section 2.13 are an integral part of this Agreement and that, without these agreements, the other party would not enter into this Agreement. The parties hereto further agree that no other party or any other Person shall be required to obtain, furnish or post any bond or similar instrument in connection with or as a condition to obtaining any remedy referred to in this Section 2.13, and each party hereto irrevocably waives any right it may have to require the obtaining, furnishing or posting of any such bond or similar instrument.

Section 2.14 Stockholder Capacity. No Person executing this Agreement who is or becomes during the term hereof a director or officer of the Company shall be deemed to make any agreement or understanding herein in his or her capacity as such director or officer. Stockholder signs solely in his, her or its capacity as the beneficial owner of the Subject Shares and nothing herein shall limit or prohibit Stockholder or any of its Representatives, in his or her capacity as an officer or director of the Company, from taking any action or failing to take any action in such capacity.

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IN WITNESS WHEREOF, the parties hereto have caused this Agreement to be duly executed by their respective authorized signatories as of the day and year first above written.

[Signature Page to Company Support and Irrevocable Proxy Agreement]