Getting from Silver City to San Antonio, while bypassing both I-10 and Del Rio is no easy feat. You might take Hwy 180 through Carlsbad and east, then cut south on 87. You'll be in a lot of desert & plains, but at least you'll be off of Interstates 10 & 20.
One way to help the issue, if you have a GPS, is to set your routing info to "avoid highways"; the GPS will look for ways to keep you off the interstates. You can also uncheck the "avoid unpaved roads" box, and your GPS may lead you down some unexpected byways. Of course, it'll take you a good bit longer to drive home, but I assume you're prepared for that. Safe travels.
When I was a college kid, I regularly plied the road from El Paso to San Antonio. In those days, they were in the process of building I-10, and there was lots of two-lane left. You can certainly make that trip faster these days, but that comes at a cost. Before it became interstate, the road from Sonora to San Antonio was not only scenic (which it still is), but a lot of fun to boot.