Are there any fly/trout streams in the area?
There is the Rio Peñasco in the area. Mesilla Fly FIshers club has a lease on one of the sections of the river - you may be able to fish it through them. McCock Ranch (I believe that is the name) has the rest of the river on their property; it is a world class fishery. They may have day spots open; there is a waiting list to join the club to fish there, they say it is that good.
There are other streams in the area that get stock. Check the game and fish site for stocking sites. For quality fishing that is open, you have to travel far north.
Thanks for the info.