I wrote up the start of a planned short timeline based on this. The ultimate goal is to have the Republican Party collapse, so some parts are going to be tilted against the Republicans.
Part One: Change in Republican Leadership
After the humiliating defeat in the 1936 elections, the trouble for the Republicans continued even before the 75th Congress began. In Oregon, Senate majority leader Charles McNary, who had been one of the major unifying figures in the party, was defeated for reelection. At the time, there was no Republican whip because there were already so few Republicans in the Senate that it was not needed. This meant there was no obvious successor for the position of majority leader. Immediately, party figures in the Senate began debating who would be McNary's replacement. Gifford Pinchot, one of the most influential members of the progressive wing of the Republicans, supported Hiram Johnson of California for the position. Meanwhile, the more conservative members of the party sought a suitable candidate, and found their man in Frederick Steiwer, McNary's fellow statesman. The struggle for the party leadership in the Senate already underscored the unraveling of the party as a whole. Things soon became even more dire for the Senate Republicans with South Dakota senator Peter Norbeck's death in December 1936. Tom Berry, elected to a third term as governor of the state in November, appointed Democrat Herbert Hitchcock to replace Norbeck, a Republican. This brought the number of Republicans in the Senate down to eight.
The contest for the Senate party leadership was brief, but emphasized the deeper divisions within the party. Hiram Johnson was a general supporter of the New Deal during Franklin Roosevelt's first term. The former Bull Moose frequently crossed the aisle to support the President's economic package, and saw and understood the federal government's role in the turnaround of the national economy. The potential nomination of Johnson saw a scandal of its own erupt within the ranks of the Republican Party, as Johnson had gone so far as to endorse Roosevelt in the 1932 and 1936 presidential elections. However, so desperate were the Republicans at the time that Hiram Johnson was seriously considered for the position. To oppose Johnson, more conservative Republicans proposed Frederick Steiwer for the position of majority leader. Steiwer, the keynote speaker at the 1936 Republican national convention, was a staunch opponent of the New Deal but, like William Borah, not a full blown conservative. Steiwer won out in the selection of the minority leadership. As with McNary's tenure as minority leader, no official Republican Whip was appointed due to the small Republican representation in the Senate.
With the heavy loss by the Republican Party in 1936 in the presidential election as well as in Congress, many prominent Republicans called for the replacement of John Hamilton, the chairman of the Republican National Committee. Hamilton was the leader of the Kansas Republican Party, and was chosen to chair the RNC after the 1936 convention and the nomination of Kansas governor Alf Landon as the 1936 presidential candidate. However, after Landon's landslide defeat and the Congressional losses that came with it, Hamilton was reconsidered for the chairmanship. Arthur Vandenberg, a former Michigan senator who was defeated in 1936, was approached to take up the Republican Committee chair after the start of the new Congress in 1937. Vandenberg had been considered for the vice presidenital nomination in 1936 in Cleveland, but declined due to the likelihood of Roosevelt's reelection. Vandenberg accepted this new offer, however, wanting to stay in the political spotlight after losing his seat. The outspoken conservative was thought to be a good choice due to his ability to work with conservative Democrats in the Senate to oppose many New Deal initiatives during Roosevelt's first term.
New positions:
Senate Republican Leader: Frederick Steiwer (OR) replaces Charles McNary (OR)
Republican National Committee Chair: Arthur Vandenberg replaces John D. W. Hamilton
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Hamilton's replacement is needed because he played a large part in building the Republican ground level organization and rebuilding the confidence in the party that allowed them to resurge as much as they did in 1938 in OTL. I wasn't quite sure who to replace him with, but Vandenberg seems like a decent choice.