I agree with your decision to avoid southwest Texas. There is really not much to see there.
When you are at the Alamo, walk over to the Post Office nearby and look at the gorgeous frescoes they have inside. They cover the history of the area in several scenes. Fortunately, one of the guards pointed them out to us or we would have missed them. The Post Office is also rather unique in its construction. It definitely reflects the architecture of the time. Plus, there are displays in the walls about some of the history of Texas. And, if you ask the guards about it, they can point out where the markers are located at the sides of the Post Office that mark the original location of the North wall of the Alamo complex.
And while you are at the Alamo you might want to visit the River Walk. We certainly enjoyed taking one of the boats around the river in downtown San Antonio. The River Walk is a very unique experience. (The river is a short walk from the Alamo.)
While you are in San Antonio you should take time to visit all five missions. They all played a part in the history of the area. They are all located about an hour's ride by horseback apart.