Have Taft beat Eisenhower for the GOP nomination in 1952, then lose to Stevenson. The GOP would be in despair ... They ran me-too candidates and lost, then ran a true conservative and lost. They might fragment, extending Democratic control of the White House and helping the Dems control Congress for years to come.
The Democrats lost six presidential elections in a row from 1860 through 1880 without despairing or fragmenting.
Anyway, if Taft lost it would only be narrowly and quite likely the Republicans would gain control of both houses of Congress in 1954 (the post-Korea recession, dissatisfaction with the results of the war, [1] the Soviet H-bomb, etc.)
[1] To be sure in OTL people were glad to see the Korean War end but not thrilled with the results. In OTL, the GOP didn't get much blame for the peace terms because of a widespread belief that Truman had made them inevitable. In this ATL, nobody but the Democrats will take the blame for the "no-win" war.)
Major political parties in the US have not given up or fragmented after bad election results since the Whigs. And in those days it was easier to supplant an old opposition party with a new one. To quote an old post of mine:
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Michael F. Holt in *The Rise and Fall of the American Whig Party* notes that some Whigs attempted to keep their spirits up after their party's 1852 defeat by saying that the victorious Democrats were bound to make some terrible mistake. As he notes, they were right about that--the Kansas-Nebraska Act shattered the Democrats in the North. The Whigs were wrong only in thinking that Democratic mistakes and disunity would automatically benefit *them.* As Holt writes (pp. 772-3):
"During the twentieth century, American electoral politics has always been organized around the same two major parties--Republicans and Democrats--in large part because the adoption of state-printed ballots in the 1890s measurably increased the difficulty of launching a third party to challenge them. Since those major parties had an automatic slot on the ballots and since the legal hurdles for other parties to get on those ballots were so high, Republicans and Democrats effectively monopolized voters' choice. During this century, therefore, the Republican party has been the only realistic alternative to the Democrats. Thus it, and not some other party, has usually benefited when voters sought to punish Democrats and to replace them in office.
"In the 1850s and for most of the nineteenth century, however, the rules of the political game encouraged rather than inhibited the creation of new parties. Instead of state-printed ballots that gave legally recognized major parties pride of place and disadvantaged other groups who sought to be listed on them, political parties distributed and printed their own ballots. As a result, it was far easier for new parties to challenge the old ones. As Whigs would learn to their dismay, therefore, politics in the 1850s was not a zero-sum game...Unlike their twentieth-century Republican successors, in sum, Whigs could not monopolize opposition to Democrats and that simple, if easily overlooked, fact more than anything else explains the death of the Whig party."
According to Holt (p. 1130, n. 24):
"For evidence that Democrats and Republicans cooperated in many states to adopt the Australian ballot during the 1890s explicitly to deny third parties access to the electorate, see McCormick, *From Realignment to Reform*, pp. 114-18; Reynold and McCormick, 'Outlawing "Treachery"'; and Argersinger, 'Place on the Ballot.'"