Guadalupe Mtns. NP is going to be your best bet for fall/winter/early spring backpacking trips nearest Midland. There's plenty to see and experience in and around the Guadalupe Mtns. Multiday trips are very possible in the Guadalupe Mtns. with creativity, but the sparsity of reliable water makes weekend overnighters the ideal trip length. This region is defined by deep canyons and high mesas. Pine, Juniper, Pinon, Oak, Maple, Mesquite- all tree varieties you can find up above the desert floor where most of the hikes originate in the Guadalupes.
You're less than 5 hours away from the Sacramento Mtns and the White Mountain Wilderness area where you can put any length of trip together. From Ruidoso, NM you can make your way into the White Mountain Wilderness and connect with the Crest Trail starting at Ski Apache (Sierra Blanca. 11,981') and make your way NE along the ridgeline. Wildfires have changed much of the landscape, but the region is slowly recovering and the trails are open and significantly cooler than the high desert below.
Midland to Gila Wilderness. 6-7hr drive and lots of remote canyons and mesas to explore. Easily one of my favorite slices of backcountry heaven. Come here if you want to escape pretty much everything and avoid the crowds. Only "downside" to the Gila now is the prevalent fire damage from a massive fire that wiped nearly the entire forest. Recovery has begun but this region will never be the same. If you do decide to head to the Gila, plan trips down along the Middle and West Fork of the Gila Wilderness where water availability is extremely reliable.
Midland to Pecos Wilderness, Jemez Mtns. 6-7hr drive. Pecos Wilderness is going to be your closest destination for big alpine mountain trips. Hit the road by 4am and you can be on the trail hiking up to an alpine lake by the days end. Worth every bit of the drive. I have friends that made the long drive from OKC to the Pecos for a trip and met up with me for beer and pizza that same afternoon in downtown Santa Fe before hitting the trail in the late afternoon.