Who was elected president defeated in re-election and ran again?

Grover Cleveland was defeated in his bid for re-election in 1888 but ran again and won in 1892. He was the only president to serve two non-consecutive terms.

Martin Van Buren came close to getting the Democratic nomination in 1844 after losing his re-election bid in 1840. In 1848 he ran on the third party, Free-Soil ticket and probably tipped the election to the Whig Zachary Taylor by taking away Democratic from Lewis Cass votes in New York.