No former president has had a permanent job in government, since William Howard Taft retired from the Supreme Court in 1930.
There has been some talk about them doing so. Bill Clinton was mentioned, not terribly seriously, as a candidate for the U.S. Senate from either Arkansas or New York. The possibility of his going on to the Supreme Court was also discussed. But, aside from questions that would probably have been resurrected about his behavior toward the judicial branch in the Paula Jones case, many thought that that job would not fit his active personality.
On the other hand, the talk about Gerald Ford becoming Ronald Reagan's running mate in 1980 was quite serious. But, frantic negotiations between their representatives, during that year's Republican National Convention, did not come to a successful conclusion.
During recent decades, a template has emerged of activities in which former presidents engage, that George W. Bush has said he will follow. That is especially true of presidents who retire relatively young. Bush is in the middle in regard to age. He is 62, putting him older than Jimmy Carter and Bill Clinton at that stage, but younger than his father, as well as Ronald Reagan.
Writing at least one book, and founding a presidential library, constitute the bare minimum. The more recent presidents have created a larger structure centered on their libraries, with names akin to "presidential center". Those centers have carried on some combination of scholarship and political/charitable activism.
George W. Bush's library and center will be built on the campus of Southern Methodist University, in Dallas. For some reason, the Texas presidents have followed that pattern. Lyndon Johnson's library is at University of Texas--Austin, and George H.W. Bush's is on the campus of Texas A&M University, in College Station.
As far as I know, none of the other presidents' libraries are as closely tied to universities. I thought I had heard that the Kennedy Library was somehow affiliated with Harvard, but their press office has confirmed to me that that is not the case. However, they sponsor the New Frontier Award, jointly with Harvard's Kennedy School of Government.
I'm reminded of the old saying: there are three ways to do anything: the right way, the wrong way, and the Texas way. For whatever reason, the Texas way is to site presidential libraries on college campuses.
Some time ago, I noted with interest that, while former members of Congress often stay in the Washington area, former presidents usually do not. I later realized that that was primarily because the former members of Congress often got jobs as lobbyists. I suppose such work would be considered unseemly for a former president.
Former presidents have either moved back to their home states, or migrated to places such as New York or Southern California.