I have never been, but I am excited to try out the open gym at Alamo Heights Gymnastics Academy for ages 0-6. Basically, they open the gym up and you let your kiddo/s go to town.
There is an entrance fee. Also, the 10:30-11:30 time slot is set by them--playtime is an hour on the nose. It is right next door to Brooks Hollow library, so if you want to make a morning of it, check out some books before heading over!