New York’s presidential primary is still four months away, but while other candidates are focusing on early-voting states like Iowa and New Hampshire, Michael Bloomberg has already begun assembling a campaign apparatus in his home state.

The billionaire former mayor of New York City has tapped two men to lead the effort: John Calvelli, executive vice president for public affairs at the Wildlife Conservation Society, and Joseph Garba, an aide to Assembly Speaker Carl Heastie, a Democrat from the Bronx. Both have taken leave...