Of the one term presidents, James Buchanan, Rutherford B. Hayes and James K. Polk had each campaigned with a promise to not seek re-election and accordingly retired from politics following their terms. Most of the other one-term presidents ran for re-election, but decided to retire from politics after their loss: George H. W. Bush, Jimmy Carter, Gerald Ford, Chester Arthur, Benjamin Harrison, Franklin Pierce, and John Adams. John Quincy Adams, Andrew Johnson and William Taft did not run for president again following their losses, but did stay active in politics (Adams joined the House of Representatives, Johnson made a failed run for a Senate seat, and Taft later served in the Supreme Court).

There are 4 US Presidents I'm aware of who intended to run for President again following the loss of their re-election: Martin Van Buren, Herbert Hoover, Grover Cleveland and Donald Trump. By the end of Hoover's first term, however, he was unpopular even within his own party and never really had a serious chance at a second run. Van Buren continued to be quite popular among Democrats after he lost his re-election in 1840, but he lost support after arguing against the immediate annexation of Texas (which most Democrats favored because it would add another slave state to the union) on the grounds that it could start an unjust war with Mexico. He still ran in 1844, and almost clenched the Democratic nomination, but ultimately that went to Polk. He ran again in 1848 as a nominee of the newly established Free Soil party, but only managed to secure 10% of the vote.

Grover Cleveland won the popular vote during his re-election bid in 1888, but Benjamin Harrison was elected president by the electoral college. Cleveland stayed popular during Harrison's term, however, and managed to beat him in the election of 1892, becoming the first and only president to be elected to non-consecutive terms.