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.
There is one tip I have learned over the years in hunting whitetails in late season snow. Do not overlook soybeans left standing on the edges of fence rows, woods, creeks, buildings wherever it might be, this is where the deer will be while the food lasts during late season. December/January will pull deer in from some distance away to feed on the easy to get to food.
I killed one of my nicest bucks off of a very small patch of soybeans that was left after rains created a...