In the US, third parties tend not to do too well. Mostly this is due to a mix of Duverger's law and the parties adapting to incorporate the minor party's aims.
But it's been argued that for a time the US was a de facto four-party system: Southern Democrats, National Democrats, Conservative Republicans, and Moderate/Liberal Republicans.
Perhaps a point in which two splinter movements emerge at the same time would make for a system in which the Democrats and Republicans are the clear big parties, but a couple of smaller parties could hover around.
There's the American Independent Party which could perhaps be a different ship for southern whites to jump into than the GOP. Yet at around the same time, you had the liberal movement (cultural liberal) emerging and you had the Koch-infused libertarians emerging which maybe could have coopted the McCarthys and other anti-war free-speech liberals.
On the other hand, in the 90s-2000s perhaps there could be a dual emergence of the Greens (like in Wilcoxchar's TL) and a more stable reform party.