best route to bypass san antonio tx heading south

From Junction, take 83 south all the way down to highway 44 and take that east over to Alice and 281. the only "bad" stretch is when you first get on 44, for about 15 miles to Encinal, after that it's a good two lane road, no problems.
Fuel up in Encinal at the Love's.

If that doesn't sound so good, take I 10 down to Loop 1604 and take that south (to the right) and go all the way around to 281. You'll have some traffic, but it's all pretty straight forward.