PHILADELPHIA—Michael Bloomberg’s campaign is planning to keep offices open and staffers employed through the November election—even if he isn’t the nominee—in six battleground states President Trump swept in 2016, according to people familiar with the matter.

The plan marks the latest wrinkle in the billionaire’s unorthodox presidential campaign, in which he entered the contest late, is skipping the early-state nominating contests and pouring part of his personal fortune into political ads.

Mr....