Wendell Wilkie had no experience in political office before running for President in 1940. OTL, he got trounced by FDR, but by a much smaller margin than Hoover or Landon and by not much bigger a margin than Dewey in 1944. Had FDR had to drop out of the race late for health reasons in favor of his running mate Henry Wallace, Willkie would have had a much better shot, especially since Wallace suffered a rather embarassing personal scandal by the standards of the day (regarding his New Age spiritual practices) during the campaign IOTL.
Alternately, Dewey was a contender for the Republican nomination in 1940, and lead the voting by a wide margin on the first ballot. At the time, his highest elected office was District Attorney for Manhattan. Dewey's support collapsed during the convention in Willkie's favor because he'd taken a cautiously noninterventionist view on WW2, while Willkie had been outspoken in advocating material aid to Britain, and the news of the Fall of France had shifted public opinion towards Willkie's position. ITTL, Dewey also takes a pro-Allied stance relatively early in the campaign, and winds up winning the Republican nomination. As per the Willkie-victory scenario, FDR still drops out of the race, giving Dewey as shot against the much weaker candidate Wallace.
William Jennings Bryan could probably have won in 1896 or 1900, and he'd never served higher than the House (his stint as Secetary of State came later under Wilson).
Apart from them, I don't see anyone else who fits the bill with a campaign-timeframe POD. With a POD a year or more before the election in question, there are no doubt a lot of opportunities, but nothing leaps to mind for me right now.