N.M. Stat. § 13-4-22
In the event any such contract shall not contain the provision above set forth in Section 1 [13-4-21 NMSA 1978] of this act, or in the event the agent so designated in such contract shall die or remove from the state of New Mexico, then and in such event, the nonresident person, partnership or foreign corporation, as the case may be, by entering into said contract shall be deemed to have named the secretary of the state of New Mexico, and his successor in office, as the true and lawful agent of such person, partnership or corporation upon whom such legal process or writs may be served in any action arising out of such contract, and when service is made upon the secretary of state in the manner hereinafter provided, such service shall have the same force and effect as though personal service had been made upon such person, partnership or corporation within the state of New Mexico.
NMS § 13-4-22