The Labor Department has promoted Molly Conway to chief of staff, where she will succeed Labor Secretary Alexander Acosta’s controversial outgoing adviser.

Conway on June 1 will take over for Nicholas Geale, who was the subject of a White House probe into accusations of mistreating employees and misleading the White House on regulatory progress. She currently serves as Acosta’s deputy chief of staff and will remain in her dual role as the acting head of the DOL’s Employment and Training...