I think the lack of replies is due to folks getting out and enjoying the wonderful weather we have been having lately here in San Antonio and thoughout Texas.
I second the previous recommendations, and offer up a few others --
Hill Country State Natural Area in Bandera (about 40+ minutes out, so just slightly beyond your desired travel time) has all sorts of spiders and lizards, if all my camping experiences there are a reliable indication. I have enough trouble identifying birds, so I'm afraid I can't name species off the top of my head. Hill Country SNA is one of the few parks I've camped at that still has "lightning bugs", for example.
Enchanted Rock State Natural Area near Fredericksburg might be a treat. Again, the place is a haven for lizards -- they love the pink granite outcrops, and the crevices, caves and crannies that accompany them.
If you really, really feel like driving, one of the parks unlike all the others in the state is
Palmetto State Park in Gonzales. It's swampy and a "whole 'nother world," and that's saying something in a state that bills itself as a "whole 'nother country".
PS: When you visit
Guadalupe River State Park, if you happen to be there on a Saturday, do take the guided
Honey Creek State Natural Area tour. It's an inexpensive donation (couple bucks) for a walking tour of the Natural Area which is located in Guadalupe River State Park and not open to the public except for tours.
Hope you enjoy your time here in San Antonio! I've called it home for nearly my entire 31 years, and Justin's been here forever.