Eisenhower at one point entertained picking former Governor of New York and two-time Republican presidential nominee Thomas E. Dewey as his successor instead of Richard Nixon, but the idea was quashed. Dewey remained a close friend of both until his death.
Former Illinois Gov. Adlai E. Stevenson tried to run for president for the third consecutive time in 1960, challenging three eventual presidential nominees. He still maintained a sizable following before being appointed as Ambassador to the United Nations.
Get these guys running against each other.
There is no way Dewey could get the GOP nomination after his 1944 and 1948 losses--whatever Ike thinks of him. Someone like Bryan could engender such loyalty from his followers that he could get nominated in 1908 despite 1896 and 1900, but Dewey was hardly the type to attract fanatical devotees. (And anyway, Bryan was only nominated in 1908 because the anti-Bryan conservatives had gotten their turn in 1904 and done even worse than him.)
However, I do see a simple way to get a Stevenson-Dewey matchup--in 1952. Have Dewey lose the 1948 nomination to Taft, and have Taft lose to Truman. In that event, Dewey (who will win re-election as governor of New York in 1950) is a very plausible candidate for the GOP nomination in 1952. (One can see his argument: "Yes, I lost in 1944, but that was in wartime, when people didn't want to swap horses in mid-stream, etc. And it was against FDR, who, whatever else you may say about him, was a master politician. Taft couldn't even beat an unpopular Harry Truman--and a fractured party--in peacetime, in an election everyone said our party should have won!") In OTL, Eisenhower ran in 1952 largely because he was worried that an isolationist (to use an oversimplified label) like Taft would win the nomination; with Dewey in the race, he may not feel the need to run. So one could certainly see a Dewey-Stevenson race in 1952. But I just don't see it in 1960--especially since for those Republicans who were to the left of Nixon (and they were probably a hopeless minority in any event) Nelson Rockefeller had emerged as their only hope.
