When Jeb Bush set out to build a political base of his own in South Florida three decades ago, he did not lack for new friends. He was a magnet for a long line of Republicans eager to be associated with him and his powerful family, including a Cuban immigrant and former corrections officer named Tony Campos. Mr. Campos ended up working as an advance man for Mr. Bush’s father, George, when he ran for president in 1988 and on the inauguration after he won. Jeb Bush was impressed enough that he recommended Mr. Campos for a job in the...