You'd need a disastrous election, both on the federal and state levels, in 1938 and 1940, for the Republicans to be well and truly finished as a major party. So you'd have to avoid the 37-38 recession, and probably have battles within the Republican Party over the party's future. Say, pro-New Deal Republicans having vary bloody and public political disputes with traditionalists, to the point where they end up demoralizing Republicans, potentially splitting the vote, or even joining the Democratic Party.
So, you'd need bad results in 38, a wave of defections and retirements, and then crucially, no Wilkie type moderate in 40. Some traditionalist has to win the nomination and choose to stay the course. Then maybe the party's organization would whither out of major party status.
Ultimately, the new hegemonic Democratic Party will not last for long. Most likely, former Republicans and such just end up joining the party, leading to an internal conflict over the party's future. In this case, what is most likely are two factions defecting from the Democrats. A more left-wing faction, closely allied with organized labor and probably a focus of entryism by socialist and communist groups, and a Southern Dixiecrat faction.