There are not too many places where you can go "10 miles off the roads". In fact you will probably have to do a lot of driving and glassing from the roads, covering ground is important. Find a few spots off the road can be helpful, but I would put more emphasis on covering ground until you find oryx or fresh sign.
Checking for crossings at the WSMR fences can be useful (oryx usually crawl under fences like antelope), but lots of other guys will also be doing that.
It wouldn't hurt to call the game wardens in Socorro, Carrizozo, and Las Cruces, in the past I've got somewhat useful info from them. There is also a small chunk of BLM land on the NE corner that requires a special access permit, so it doesn't always get hit that hard. It is also a pain in the a$$ to get to as they make you drive way out of your way to get to it, and the road really sucks in a couple of spots. Not even sure if this is still the case, but it might make a backup plan. I think you have to contact BLM office at Valley of Fires. Last time I went in there (~5 years ago?) I passed up a long shot at a broken horned cow (almost regretted doing that). I guess that brings up a point; if you get a shot, any shot, take it!
