John Tyler renominated by the Democrats in 1863. Or, Polk decided to go for a second term.
A (reasonably realistic dystopian) conservativewank timeline which starts way back with Grant deciding to run for the third time leading to progressives in office at the time of the Great Depression. Of course, I don't have the knowledge of the time, the Congress, Grant, or anyone really to write it.
A (reasonably realistic dystopian) progressivewank timeline which starts when a far worse Great Depression results in a slew of third-party candidates and split tickets in 1932 the end result of which sends Hoover back to the White House for a second term until he is assassinated shortly after inauguration, Charles Curtis dies of a heart attack earlier then OTL, and, well... I don't have the knowledge of the time, the Congress, or Stimson for it.