Take 285 a few miles south from Carlsbad and take 137 towards Dog Canyon campground at Guadalupe Mountains National Park. You will drive through the small town of Queen (last stop for food if the one place is still open).
From here I can't remember the exact cuttoff but you will want to head to 5 Points Vista. Beautiful view from there. Google should be able to direct you there.
https://www.fs.usda.gov/recarea/lincoln/recarea/?recid=79402
From there you keep heading south on a dirt road and then you will find quite a few 4x4 roads, one of them leading to Dark Canyon Lookout Observation Site.
https://www.fs.usda.gov/recarea/lincoln/recarea/?recid=79402
Anything with two dotted lines are mostly 4x4 roads, not terribly difficult but pretty rough. Great scenery.
The Camp Wilderness Ridge road shows to end at trail 45 but in fact it continues on to the Texas border where you can then hike a portion of the Permian Reef Trail for a spectacular view of McKittrick Canyon.
Here is a YouTube slide show, but don't watch it if you want this new place to be a surprise.
https://www.youtube.com/watch?v=8EZqYqUx-Os
Also, just a quick side trip, you may want to go by and check out Sitting Bull Falls for a few minutes. Mildly interesting.
Also, if you have never seen the Organ Needles near Las Cruces, I recommend driving over to the Aguirre Springs campground for a spectacular view of the Organs and some hiking trails. If you drive around to the west side (Las Cruces side of the Organs) there is a dirt road that parallels the Organ Needles and with snow caps it is one of the most spectacular ranges in the U.S.