Perhaps your only chance to play a private course like the one at the U of Texas Golf Club, unless you know a member, is to sign up to play on September 26, when dozens of ClubCorp private courses permit outside play for the benefit of charity.
My friends at Golf Vacation Insider reminded me today in an email that dozens of private golf clubs in the U.S. are available for play by non-members for one day, September 26. ClubCorp, which is organizing the event, calls it the "largest one-day charity golf and dining event" in the world. I recognize a few favorites I've played, like the University of Texas Club (Bechtol Russell Design) and Hills of Lakeway (Jack Nicklaus), both in the Austin area, and Woodside Plantation in Aiken, SC (45 holes, Rees Jones & Bob Cupp).
Prices range up to $150 for golf and dinner. An online auction is also part of the event. For the full list of golf courses and dining clubs, plus a sign-up form, click here.