>>1949003NM 152 has the best trails in Gila, I think. It's a bit further east than most of the forest, but that works towards its advantage since it's FAR from the Cliff Dwellings and the Catwalk, the two biggest "tourist" attractions in the forest, so you hardly see people. The trail to Hillsboro Peak, Railroad Canyon, and Rabb Park are some of my personal favorites.
NM-152 (pictured) is also one of the most scenic and beautiful roads in America so there's that, very enjoyable drive.