The only way Bill could become President again would be if he was Speaker of the House and both the Presidency and Vice Presidency became vacant at the same time. I believe the House of Representatives is permitted to select as Speaker someone who isn't a member of the body, but I don't that has ever actually happened.
If the speaker of the House (or the president pro tempore of the Senate or a Cabinet member) is ineligible to be president, then the line of succession would skip to the person below him. For example, Henry Kissinger could not have succeeded to the presidency when he was secretary of state.