Lake Travis is another option in Austin. Surrounded by more of a residential area there are a few public beach spots to get into the water. A popular lake for sailing, boating and swimming. There are a few state parks surrounding the Lake. The water levels used to be incredibly low but have come back in the last couple years making it a great spot for a dip.
The last is a gym owned quarry with a triathlete swim course built into the lake (plus SUP if you're interested!) It's the Quarry Lake location of Pure Austin fitness. They offer day passes for visitors so you can hit the gym and go for a swim in either the pool or the lake.
Hope that helps!
Stevie (@steviepeters) and Dean (@dc_fit_lab)