If you're visiting Alamogordo, you will likely take some time to visit the Space Museum... it is a nice diversion. While there, off the back of the parking lot is a well developed hiking trail that winds up and down and around through the desert scrub, arriving within half a mile up at a nice overlook with a graphic display that talks about the mortar holes located in the surrounding washes that are the basis for the name of the trail: they were called Indian Wells by the early locals. The trail also includes some name tagged flora along it's well demarked route.
Once at the signed highpoint,the official end of the trail, the path obviously continues on to the north and east, though not so well maintained. It winds up and down and around the washes below the towering ridgeline to the north. If you are really good at route finding, you can follow this faint trail all the way over to connect with Mineral Springs trail along it's old jeep route section. If you are not so good, you will find yourself wandering in the low, sticky cacti and mesquite scrub. A good approach to reconnect with the lost trail is to walk ahead 20-30 feet, remaining aware of your last point, then turn and walk a circle oriented around that point. You will likely recut the lost trail, and if not, you will cut back into the trail where you left it walking in. You may expand this looping search out to 100 feet if you feel confident about your orientation to your pivot point. If all else fails, simply walk down one of the many washes; they all terminate back down on Scenic Drive near Indian Wells Road.
Note: parking lot for the Space Museum is maintained with hours currently 8am to 6pm