Upon the expiration of their term of office, all persons who have served as President or Vice President shall be appointed as At-large members of the Senate. Their term of office shall continue for life. No such At-large members shall not be eligible for appointment for offices that stand in the line of Presidential succession.
(Our Presidents and VPs are some of the few true statesmen that we have, and we are really under-utilizing them. The Senate would be a good place for them. Throwing in a few unelected members of high stature and unbeholden to any particular state would also change the dynamic of the Senate quite a bit. We might see a bit less partisan polarization and political game playing.)