One of the best places around Austin is Lost Maples State Recreation Area near Vanderpool in the Hill Country. Well within your driving parameters and I have hung a hammock there many times. Springtime and the Mountain Laurels will be in bloom (they cover the place). Most of the backcountry sites (B, E, F and H) have a very steep climb to get to and keep out the large ice chest toting' group out (they tend to congregate around site "C" which is very popular for fair weather campers. Since it is such a popular park, you need to make reservations several weeks in advance as they fill up quickly.
I am a 28 year-old trapped in a 55 year-old body.