I've not had much experience with the Gila. I know the Lincoln forest near Ruidoso/Cloudcroft has a healthy bear population, but again, I'm not really familiar with the area. The Ranger district or the area game warden could probably help you out.
If you're coming from El Paso, you may consider the San Mateo mountains, unit 17. They are only an hour or so north of Cruces. I've done a lot of deer hunting in there and I see bears about every trip. Best bet would be to get up high towards Springtime campground or the Apache Kid wilderness where the Ponderosa pines grow, get on a high spot and glass the openings and little meadows. Spot and stalk opportunities abound.
Not the biggest bears in the world, but fun regardless.
Every year many hunters and outdoorsman and women come out west from the midwest and east coast to hunt the prized mulies and elk. One topic that comes up often is altitude sickness. My advice for flatlanders is to get into the best possible shape. Start months before your hunt, usually really ramping up my cardio around March or April.
I run 5-10 miles 3 times a week, and also go for walks carrying my pack with about 50lbs to simulate what could be on my back. Another useful tip is to drink A...