Here are some random good memories of surface roads south of I-10 and west of Corpus Christi. Don’t know what if anything the pandemic has changed (I fondly hope, not much).
(Some of these echo Mr. Noodly’s and others' suggestions — nice pics, Mr. N!!)
I’ve taken Route 90 from Uvalde to Van Horn (where it ends and hooks back up with I-10) a couple of times and enjoyed the drive a lot. To me it seemed very scenic and western-feeling. Some stops on the route:
Brackettville has the Black Seminole Indian Scouts cemetery, which is very evocative if you like history. I think they even have a museum now (
https://www.seminolecemeteryassociation.com/black-seminoles-a-historical-overview.html).
There is also a museum and a pretty nifty spring-fed swimming pool at Fort Clark Springs, which is nearby (
https://www.fortclark.com/). The fort is now a residential community. When I was there, the pool and walking trails were open to the public, and I think they still are. The pool was out of service but I think they’ve fixed it up since then. Very pretty.
Keep heading west and you come to the Amistad National Recreation Area. Lake Amistad is a reservoir that is shared by Mexico and the United States (hence the name — “amistad” = friendship). You can drive out to the dam; I can’t remember whether you can drive /onto/ it but you can definitely walk out onto it. It has both national flags and the view is impressive. When I went, I had a pretty scruffy-looking car with all my earthly possessions in the back and I got stopped by some kind of border cop, I don’t remember what agency, who was a bit suspicious at first but plenty pleasant. Still, if you go out there make sure you have your paperwork together first. Per Google Maps there is a campground at the Recreation Area.
Heading west, you pass a couple of other parks (Seminole Canyon and White Shaman; both have petroglyphs, I think; I haven’t been). The bridge over the Pecos River is fun to see. In Langtry, there’s the Judge Roy Bean Museum, which is worth at least a quick stop. According to Google, there is even a Law West of the Pecos Campground.
There’s an RV park in Marathon, which is also a cute little town with, IIRC, some decent restaurants. There’s a county park just south of Marathon that has swimming and I want to say something built by the CCC but I could be wrong. It might be called Fort Peña Colorado Park or Post Park. A short but scenic drive.
Heading west from there you would hit Alpine, a university town with lots of amenities, and then Marfa, which is famous for art and what’s the other thing? mysterious lights?
Heading south from Marathon or Alpine you can get to Big Bend National Park, which is supposed to be awesome.
Heading south from Marfa you can get to Presidio — a small US border town — which, if you like to take long drives to remote places just because you can, is fun, but make sure your vehicle is in good shape because it really is God’s elbow room down there. When I went, I just had lunch and walked around a bit and came back; I couldn’t really say what else there is there. One of the ranches I drove past on the way had camels (OK, I’m easily amused, but it was a kick).
Heading north from Marfa or Alpine towards I-10 you would reach Balmorrhea State Park, which also has a nice old spring-fed swimming pool, which I think has also recently been renovated. It’s a nice place to walk around, too.
Otherwise, you can get about another hour’s drive on Route 90 before it T-bones with I-10 at Van Horn. At that point, I-10 is so close to the border there really isn’t a more southerly route until you get to the other side of El Paso.
I knew people who routinely took Route 9 from El Paso to Columbus, NM, famous because of Pancho Villa etc. Right across the border from Columbus is Palomas, Chihuahua. It’s easy to park on the US side and walk over. Just a couple of blocks in, there is a very gringo-friendly restaurant and store called the Pink Store. I used to live in Deming and go down there for lunch sometimes. The food is great, so is the customer service, and they have live music at lunch time (or they used to anyway). If the musicians are there, please ask them to play Viva Chihuahua, and tip heavily
The tschotchkes in the store are pretty decent, too -- they buy from artisans all over Mexico.
Back in New Mexico, outside of Deming (just north of Columbus/Palomas and right on I-10), Rockhound State Park is great. West of there I don’t know anything about the surface roads, alas.
If, on the other hand, you decide to jump back on I-10 from Corpus Christi, I only have one suggestion: Los Reyes Café in Smiley, Texas, which I would say (hope it’s still open!) is a true best-in-class for inexpensive Mexican restaurants.
Enjoy your trip!