We visited several areas within Organ Mountains Desert Peaks National Monument, Chihuahuan Desert Nature Park and also within the restricted USDA's Jornada Experimental Range and similarly restricted lands belonging to NMSU's Chihuahuan Desert Rangeland Research Center. Locked gates block vehicle access to entire northern half of the range which is controlled by these two publicly funded entities, so travel is entirely on foot.
I'll let you make up your mind on whether you want to participate in a little civil disobedience, just remember to respect the land, if not the policies of the institutions that control it. On the first trip we parked near the locked gate on the west side of the Chihuahuan Desert Nature Park. REMEMBER, THEY LOCK THE EAST GATE AT 5:00 PM
On the second visit we used one of the two dirt roads off of Jornada Road that are part of the Monument, and then used the very rough powerline road to access the areas on the north and northeast side of Doña Ana Peak. High clearance and four wheel drive are strongly recommended.