CheckMySwag
According to the current map booklet for Texas public hunting lands there are 13 public hunt areas offering feral hog opportunities in Region 6 - Austin/Waco.
None of them are in Travis County (Austin), but all of them are within an hour or two of drive time from Austin.
The closest appears to be at Granger WMA, which is only about 30 miles from Austin, just north of Taylor.
You will have to pay $48 for an Annual Public Hunting Permit to take advantage of these opportunities, but it's well worth it.
Upon further review:
Granger is open for hog hunting year around, but it's ARCHERY ONLY.
Most of the other 13 are "during dove season, only". Too late for that!
Best bets are limited to only 2 areas, for any legal means (rifle, shotgun, pistol, muzzle loader, etc.) One of them is for Jan, Feb, and March, while the other is for March and April.
Buy the permit, and you will get a 142 page book of maps and specific instructions on what, when, where, how, etc., covering almost 900,000 acres (that's nine tenths of a MILLION) acres of public hunting lands. Plus several National Forests administered by the Feds. No permit needed, just a Texas hunting license.