The most obvious answer here is John Quincy Adams, who was president from 1825 to 1829; two years after losing his bid for reelection to Andrew Jackson, Adams was elected to a U.S. House seat, which he represented from 1831 until his death in 1848. In his long second career in Congress, Adams became famous as an anti-slavery agitator, especially through his attempts to present anti-slavery petitions to Congress despite the House's "gag rule" against adopting them.
But that's as close to common knowledge as this question gets. Aside from former presidents who later tried to get re-elected, including Martin Van Buren, Theodore Roosevelt, and (successfully) Grover Cleveland, there are a few more interesting examples here, too.
Interestingly, most of the presidents who accepted or pursued lower office after being president were presidents who had never been elected to the office, but succeeded to it after being vice president.
Andrew Johnson, who succeeded Abraham Lincoln as president, and then was impeached and nearly removed from office, never lost his vigor for politics, and campaigned for election to both the U.S. House and Senate. After several failed attempts, the Tennessee Legislature finally elected him as a U.S. Senator again in 1875. "I’d rather have this information than to learn that I had been elected President of the United States." Johnson said. "Thank God for the vindication." He would die a few months later.
Most ignominiously, President John "Tyler Too" Tyler, the first vice president to assume the presidency, would be elected in 1861 to the House of Representatives — the Confederate House of Representatives. But Tyler died before the new chamber met. UPDATE: u/seccessionisillegal corrected me: Tyler did in fact serve in the Confederate House after receiving a provisional appointment to the body; he was then formally elected and died before the newly elected body met.
Gerald Ford took over the presidency after Richard Nixon's resignation, and then lost his bid for a full term to Jimmy Carter in 1976. In 1980, Ronald Reagan won the Republican nomination. Reagan had unsuccessfully challenged Ford for the GOP nomination in 1976, but in 1980, he can very close to picking ex-President Ford as his vice presidential running mate.
Reagan's foreign policy advisor Richard Allen claims to have been with Reagan throughout the hours at the Republican National Convention when the deal with Ford fell through and Reagan picked his primary rival George Bush instead. "Reagan’s selection of Bush in Detroit represented a turnabout within six hours," Allen wrote. "It came only when the negotiations with Ford, having taken on a life of their own, appeared to have reached an impasse."
As part of the negotiations for Ford to accept the vice-presidency, he demanded significant concessions in return, including the right to pick certain cabinet officers (including Henry Kissinger as Secretary of State) — a degree of power for the vice president that some people dubbed a "co-presidency." Ford was willing to return to being vice-president again after having held the top job, but ultimately Reagan turned away from the deal and picked Bush instead.
SOURCES: