Cairn Trail
The trail head for Cairn Loop Trail is situated at a small pull-off on Highway 51 outside of Elephant Butte, not far from the dam. It doesn't appear to be interesting at first, being littered with broken glass pieces from thoughtless partyers, but that ends after perhaps 100-200m. It then enters the hills. At first there are no cairns, causing the author speculate as to why it was named so, but after about 2/3 mile, cairns appear regularly. The trail ascends up out of the valley and offers some very nice views of the surrounding area. Take note that at the southern tip of the loop (see photo on the interactive map), the trail cuts sharply to the right. The author accidentally proceeded toward a somewhat steep descent into an arroyo before realizing he had gotten off track. It's always good to have a gps track available for reference!
The return leg of the loop is a pleasant meandering through a couple of the canyons interspersed in the area. The trail then proceeds back uphill from an arroyo leading to NM51 and returns to the pull-off at the trail head. It is quite isolated and it is unlikely you will encounter other hikers.