If you try New Mexico now, it most likely will be a private land hunt, as all the lottery tags have been allocated.
There are several Texas private ranches that offer elk hunts, just scope out the EXOTICS category.
I used Brandon Gaudeli a few years ago in New Mexico -http://gilaguide.com/
If you go to his website and the GALLERY tab, that's me in the black shirt holding the elk head of my 328" bull!
You might check out Mulie Mike, but I think his elk hunts are already booked.