San Antonio native here. The closest nature hiking is a park called Government Canyon. Never been as it was developed after I moved to Japan.
Lost Maples and the backroads between the towns of Bandera, Medina, Vanderpool, Utopia, Concan, Leaky, and Kerrville can make for a nice winding drive thru the Hill Country, but Lost Maples is only "special" in the Fall when the colors are out.
Enchanted Rock might be your best bet for an interesting daytrip though takes 10x longer to get there than it does to do it! Big pimple of pink granite that takes about 15min to summit. Nice views from the top and the nearby town of Fredericksburg is one of the nicest Hill Country towns for lunch and antique shopping.
Keep in mind the area is in a severe multi-year drought, and many of the scenic natural water features like rivers, lakes etc, are in pretty sad shape. May days can sometimes be 100F or more and of course it is peak rattlesnake breeding season so they will need extra caution when out in the sticks.
That said, strolling or biking the San Antonio River is not a bad way to spend a day. The city has extended the path to ~15 miles and I think the development project was nearly finished when I was there last Summer. You can read more about it here,