AUSTIN, Texas—A federal appeals court will hold a new hearing on whether a Texas voter ID law has discriminatory effects on minorities—a potential blow to the Obama administration’s efforts to fight new ballot-box restrictions passed by conservative legislatures around the country.

A three-judge panel of the 5th U.S. Circuit Court of Appeals in New Orleans ruled in August that the 2011 Texas law requiring 14.6 million registered voters to show picture identification at the polls violates parts of the federal Voting Rights...