Rock Hound State park by Demming NM, is a pretty good spot. There are also many places around the area close by. Silver City has a few areas that are BLM or Forest service with some state areas. You can also do some hounding down around Hatchita area. Locate some of the clubs in the area online they will be of help also. Arizona has some designated BLM rock hound areas by Duncan and Safford areas.
Glad to help if you head over into Arizona or the area.