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.
Let me start by telling you the story of two Johns.
John number one went on a day hike with a couple of friends in the Cascade Mountains about an hour from Seattle. He somehow got separated from his buddies and became lost. John was dressed in casual cloths for a warm weather day hike and had nothing but a lunch with him. He spent three days in those mountains and luckily made it out to tell the story.
John number two, a friend of John number one, was much better prepared when...