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.
I've spent countless hours either on my hands and knees or bent over, messing up my back, looking for blood. One such instance I remember most was a hog hunt a few years ago in Mississippi. I had never shot a hog so when I poked a hole in this guy; I was determined to find him. Wearing snake leggings, I crawled on my hands and knees through the briars and the brambles and bushes where a rabbit wouldn't go. I found a spot of blood here and there but never the pig. And lucky for me, I didn't find...