Tennessee officials were unable to put Christa Gail Pike to death Wednesday for a 1995 murder after administering two doses of a lethal drug.
Key portions of the project could be complete within months.
The members' case challenging the Pentagon ban goes to trial in January.
Democrats in 23 states and the District of Columbia sued to block the order.
Supreme Court Chief Justice John Roberts is allowing the White House to continue construction on President Donald Trump's $400 million ballroom project for now.