President Donald Trump will have to renominate several picks for posts in the Labor Department and other agencies, or appoint new nominees.

The Senate wrapped up the 115th Congress with votes on some Trump nominees, but without advancing a reported deal to confirm a package of labor-related picks. That means the president will have to renominate his selections, restarting the Senate confirmation process for the 116th Congress, which begins Jan. 3.

More than 150 nominees were awaiting Senate confirmation as of Jan. 2, including federal judges. The Senate typically concludes its term with an end-of-year unanimous consent to confirm several...