Dewey wins in 1948, eeks out a slim reelection in 1952 because No Wartime POTUS Ever Defeated (Korea, natch) and by 1956 the Republicans are looking pretty damn shaky, mainly thanks to him failing to forge a viable thirdway between Taftite non-interventionism and George Marshall-style internationalism (at least that's what everyone would think, what with the expectations that he'd have had invested in him in those heady days of 1949).
In 1956 the GOP decides to run the old general against whomever gets a very decent looking Democratic nomination that year. And, heck, maybe the 1958 recession has been brought forward two years, thanks to savage defence cuts in the wake of the Korean armistice. Total hail mary pass.
But why Eisenhower would accept such a crappy role at that age, I dunno. Fear of a McCarthyite getting the Republican nomination?
(Surely this broad scenario has been done here before.)