The highest point in Texas provides a perfect wintertime hike. You’ll spend a full day hiking 4+ miles, one way, to the top of Guadalupe Peak, and you’ll gain 3,000 feet of elevation in the process. Sure, it could get cold here, and it could snow, but most of the time your biggest concern should be wind at the peak, not precipitation (hurricane-force winds are possible). And if it does snow, just drive up the road to Carlsbad Caverns and spend your time underground.
Guadalupe Mountains National Park is located about 110 miles east of El Paso on US 62. The hike begins at the Pine Springs campground, behind the visitor center.