For a nonresident with a high percentage chance to actually draw out on a bull tag in New Mexico you can look at units 51 and 53. You should check the success rate on actually killing an animal. I was going to go but decided to go to Utah instead.
If you want to hunt cow elk, Utah is nearly 100% chance to draw for a nonresident and a spike elk tag is over the counter.
If you want to hunt trophy elk
don't pay any attention to me. I just like to go and hunt elk of any kind.