I think the best way is just to mirror the developments in Europe at the turn of the 20th century: An upcoming socialist movement splits the progressive party into a social-democratic populist party and a libertarian trade oriented one. Of course, the big difference with the US is that in Europe, the traditional workers parties only became possible when the various countries established universal suffrage, which the US pretty much had since it's founding days (or since 1865 if you bring in the slavery issue...)
Still, there is something to be said with the argument that the third party was originally the second one. It just got upstaged by the workers party when that one gained traction. So the best way for the US to have a three party system would be to have the new class of factory workers in the new megacities repeatedly being overlooked by the two traditionalist parties until they start forming a party on their own.
One could make a snarky comment that given how it is now somehow plausible for both Bloomberg and Sanders to run in the same party's primary and nobody bats an eye, the Democrats are prone for some kind of Social Democrat vs Neoliberal Democrat split somewhat like early 20th Century Europe. NeverTrumpers and moderate republicans proceed to join the neoliberal democrats, and thus we have Social Democrat, Neoliberal Globalist, and Populist-Nationalist parties.
But sticking with the broader idea of "third party displaces second party but the displaced party stubbornly sticks around" I can think of three 20th century prospects.
1) Roosevelt's Progressives ultimately replace the Republicans and the GOP just sort of hangs around afterwards
2) The GOP is utterly destroyed in the 1930s politically. The Democrats proceed to split between the liberal and conservative factions, and the GOP just sort of hangs around in New England and a few other scattered congressional districts.
3) In the 1960s-1970s the Conservative movement forms its own party after continually being blocked from power and influence in the GOP. They proceed to replace the GOP as the second party, and the Republicans end up as a minor congressional presence.
It's actually somewhat fascinating. There's often discussion of the Nolan Chart (a chart with two axes, economically libertarian vs authoritarian and socially liberal vs socially conservative) but in practical political terms it seems a triangle is more accurate: Conservative, Liberal, and Socialist.