Nevermind.... It seems NM fish&game only allow 6% of draw licenses are awarded to nonresidents applying without a New Mexico registered outfitter. That is a very small percentage.
I haven’t hunted there but was told from NM taxidermist that non resident Elk and Pronghorn are pretty easy to get. If you want to drive further Oregon has Guaranteed Outfiter tags for Elk and Deer. A little pricey but afforadable.
Then cross check with the BLM maps. Hunting on BLM is generally free. It can be rugged and remote, so cross check with local areas unless you have a camper.