"Whoa, whoa. Slow down there, maestro. There is a NEW Mexico?" — Mr. Burns
Oh, Mr. Burns, Mr. Burns, Mr. Burns. Of course, there is a NEW Mexico. The term was first coined in 1563. The first colonist attempt to settle it happened in 1589, by a certain crypto-Jew.
Meet Gaspar Castano de Sosa: colonist, explorer, slaver, Jew. Alright, sort of a Jew. He came from a family of conversos and might have practiced Judaism in secret. We don't know for sure. He was pals with previously profiled Luis de Carabajal. You remember Carabajal, don't you? Also a colonist, explorer, slaver, Jew...
In any case, after Carabajal's arrest for "Judizing", Castano set north to establish a colony in the mystical lands of NEW Mexico. He wasn't exactly successful. The promised deserts threw him for a loop with a cold winter. His men revolted. Natives attacked him. Oh, and a convoy was sent from Mexico to arrest him. Castano was brought back, trialed, and sentenced to exile in the Philippines... pretty much as far from NEW Mexico as one can get.
After Castano's failure, colonists stayed away until 1598, when the Province of NEW Mexico was established. Somebody please update Mr. Burns!