WASHINGTON—Chief of Staff John Kelly over the past five months has imposed discipline and rigorous protocols on a freewheeling White House. But President Donald Trump has found the loopholes.

The president on occasion has called White House aides to the private residence in the evening, where he makes assignments and asks them not tell Mr. Kelly about the plans, according to several people familiar with the matter. At least once, aides have declined to carry out the requested task so as not to run afoul of Mr. Kelly, one of...