The relevant section of the 22nd Amendment reads:
Section 1. No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once.The key word is "elected" and the key phrase is "elected to the office of the President." No person shall be elected for President more than twice. The text says nothing about whether a former two-term-elected President can serve as President. Words matter. If Congress had intended that no former President could serve more than two terms, the writers could have said that. Or they could have said that no person shall be President for more than eight years. That would have been clear enough. Bill Clinton could be elected as VP and George W. Bush to Speaker of the House (or Secretary of ____, etc.). Both could then become President legitimately via lines of succession though neither could be elected again once he served out his term.