I have done quite a bit of riding down south in the past few years. The Guadalupe Rim road is nice, and it looks like there are lots of roads in the forest and BLM to explore around there.
Tons of ATV trails south of Cloudcroft. There is also a nice pretty easy North/South road along the eastern edge of the Mescalero Rez.
TorC has become a mecca of riding the last several years - Palomas Gap, Chloride Creek, Monticello Canyon, Mt Withington.
Lots of riding on Forest and BLM land around Socorro - east, south and west.
There is a fun network of roads around Kilbourne Hole west of I-25 between Los Cruces and El Paso. Some of the roads to the west get difficult due to lava rock.
There are a lot of tracks posted on DualSportMaps.com.
Also nearly all the roads show up on the NM Benchmark Rec Maps. These maps also show land ownership, and the roads on BLM and forest land are usually open. The minor roads on private land usually have locked gates.