AUSTIN, Texas—When Texas hired football coach Mack Brown a decade ago, it was the result of a high-stakes national coaching search to find just the right man to lead one of college football’s most prestigious programs.

Texas decided it didn’t want to go through that again when Brown retires, whenever that day comes. The Longhorns believe they already have the best guy for the job on the payroll.

Brown and Texas athletic director DeLoss Dodds announced Tuesday that first-year defensive coordinator Will Muschamp will be signed to a new long-term deal that not only keeps h...

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