I live about 200 miles (which is not far at all in this part of the world) southeast of El Paso, in the Big Bend region of Texas. I've been here for 28 years now and have been lucky enough to have friends who own ranches or work on ranches and allow me to hunt on their property. At times in the past, I have also helped with guiding hunts out here. I have killed whitetail (including coues deer), mule deer, auodad, javelina, mountain lions, coyotes and feral hogs. We also have elk here, but I have not personally killed one. Unfortunately, however, if I did not have personal contacts who invited me to hunt on their land, I would likely not have had any of these opportunities as most hunts in this area go for big dollars. If you have the ability to pay, I'm sure it wouldn't be too difficult to line something up, but public hunting opportunities are rare. There are a few drawn hunts on state land through Texas Parks and Wildlife Dept., but you must register for the drawings well ahead of hunting season.
I have done some bird hunting in New Mexico, but I have never done any big game hunting there. I'm not very familiar with the system in New Mexico, but I do know there is a lot of BLM land there which offers public hunting opportunities. You might want to inquire with locals around El Paso to see if they can give you the scoop on how to get in on some of that. I'm sorry I can't give you anything more encouraging, but I know a lot of local guys who simply can't hunt around here due to the unavailability of public opportunities and the high prices of purchasing hunts. As I said, I've been here for 28 years now and it took me several years before I linked up with the right people in order to get access to land to hunt on. If you seek out the right folks, you might get lucky and establish some contacts who can get you on the land. Good luck with it!