Short one before I can think of anything better:
A. Eisenhower has a Heart attack in September or October of 1956, raising serious questions about the President's health, and whether the country wants Richard Nixon as President. Alternatively, he can have a stroke like Wilson (he had one in 1957, but wasn't crippled by it) and electing Eisenhower is effectively electing a vegetable and shadow President Nixon, if the press finds out.
B. Joe McCarthy dies in December of 1953, and the Red Scare continues under much saner figures, such as Senator Pat McCarran (D-Nevada), with McCarthy as a martyr of the cause; Eisenhower is attacked from the right and the left, and a more anti-communist candidate in 1956 could be a possibility, but only if, for example, a recession like the one of 1958 (a very good year for the Democrats) allows for Stevenson to grow, maybe with Kennedy as his VP candidate.
Hmmm...I need more time for something better
But back with the first idea...
A Heart attack or stroke in October of 1956, if it arrives at the same time as the Suez and Hungary Crisis, might be enough to convince the public that they need a strong and healthy president in office.
Combine that with VP Kennedy (who unlike Kefauver, was a war hawk, voted with Southern Senators in 1957 and had a chance of carrying his own home state while mobilizing the catholic vote)