In March 1968, faced with the turmoil over the Vietnam War and challenges for the nomination from Senator Robert F. Kennedy and Senator Eugene J. McCarthy, who had stung him in the New Hampshire primary, Johnson announced that he was withdrawing from the race. His justification was that he had higher goals, like trying to negotiate peace in Vietnam and obtaining a tax surcharge from Congress, that would require his complete attention. For years afterward, Johnson was a tainted figure in American politics, condemned by the left and the right. Yet the groundbreaking programs he put on the books remain.

At the height of the conservative revolution in 1985, the spokesman for President Ronald Reagan’s budget bureau, Edwin L. Dale Jr. stated, “I wouldn’t quarrel that Medicare’s been a success. In fact, much of the social safety net was accomplished during the Great Society, and whatever gets trimmed, the safety net will remain intact.”

Other one-term presidents left records that were not as sweeping as Polk’s or Johnson’s, but they still persuaded Congress to enact policies with lasting significance. President George Bush was responsible for a sea change with passage of the Americans With Disabilities Act of 1990, legislation that created a huge regulatory structure to protect and assist those who were physically disabled.

Herbert C. Hoover created the Reconstruction Finance Corporation, which helped temporarily stabilize the credit system by lending money to banks and businesses, a precedent that F.D.R. built on. The one-termer William Howard Taft, overshadowed by the presidents who bookended him — Theodore Roosevelt and Woodrow Wilson — helped strengthen regulations on big business.

Sometimes one-term presidents have been able to influence public debate for years to come. Jimmy Carter was one such president. Faced with economic stagflation and an energy crisis, Mr. Carter lobbied Congress to pass energy conservation legislation. In a televised address early in his presidency in 1977, Mr. Carter explained that good policy would require “painful” sacrifices. He implored Americans to change their consumption patterns and called for insulating homes and using solar energy. Although some advisers warned that this would be unpopular, he did not desist. Congress passed only a watered-down version of what he wanted.

Mr. Carter continued to tell Americans exactly what they didn’t want to hear. But on many issues, he was right, and his vision continues to resonate. He was similarly influential with human rights, a foreign policy that initially earned him considerable criticism from Republicans, though in later decades many of them embraced his rhetoric.

Other one-term presidents have made difficult policy choices with harsh political consequences. In 1990, after the supply-side tax cuts ran up an astronomical deficit, the first President Bush reluctantly agreed to a deficit-reduction package that included tax increases. With this act, he broke his campaign promise: “Read my lips: No new taxes.” But he was willing to take the political heat. Newt Gingrich never forgave him. And Patrick J. Buchanan ran against him from the right in the primaries.