I've been deer hunting in that country. We drove in from the east via state roads, then county and finally to a dead end on forest service road. The last stretch was serious 4wd trail that ended by driving several miles up a stream. Our hunting camp was at the edge of the Gila Wilderness area.
On another trip into the Gila out of Silver City, NM, we were driving on a forest service road and saw a cliff dwelling about a half mile hike away so climbed to the foot of the cliff then up the face into the ruins using a steel cable that dropped down. Once into the ruin, it was really just a hollow in the cliff with deep dust. It turned out that part of the cliff face had collapsed leaving a rugged path to the top of the ridge and judging from the sign in the cliff dwelling, there was a pretty good chance that mountain lion had it on their list of hang out spots.
Me to. I have been visiting/camping/hunting that part of the country since around 1951. I have relatives from Silver City to Alma.