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.
Don't assume that when fall comes, or hunting season, that the big bucks, and big bulls leave. Whitetail deer bucks especially, simply hide. And they hide very well. Mule deer bucks may leave a little. Bull elk may leave, but don't assume they do. Don't assume any of them do. I mean - where would they leave to? (Just another place where, for the most part, they would be hunted.) These big animals have the ability to hide in very small, very obscure places. So, embrace the fact (...