The major problem you have with DCA, is the perimeter rule, that prohibits most flights to AUS or SAT. Both DFW and IAH fall within the perimeter range. I'm not aware of any exceptions for these airports (I know of some for PHX, LAX, and SAN).

Otherwise, I don't think it really matters if you fly to DFW/IAH and connect from there to SAT, or fly first to another hub, such as CLT or ATL. Generally when I visit family in PA, I'll just take the AUS-PHL nonstop, but I have done AUS-CLT-ABE and AUS-CLT-AVP to get closer.

If it isn't too difficult for you to get to IAD, I'd just fly nonstop IAD-SAT. Less chance of weather along the way.