There are FPTP systems with viable third parties. Britain, for example, in the Mother of Parliaments.
What you need is a political force distinct enough in a particular region that isn't served well by the major parties. The classic example is the South, due to its long and troubled history (or perception of the same, racism and segregation were hardly unique to the South, but only the South seceded over race issues). The Dixiecrats, of course, or the AIP could be good starting points there.
But there are other possibilities. Perhaps with a later WW1 (or one without Woodrow Wilson at the helm) a stronger Socialist Party might ensue in places in the Mountain West or the Upper Midwest (who at that time already had a reputation for radicalism, in other ways, like with free silver or with Populism generally). Or one of the many fringe ideologies of the Great Depression finds roots among the farmers (Social Credit, for example.)
Will it necessarily lead to third party presidential victories or kingmaker scenarios? Unlikely, but they could do so at the Congressional and state house levels (and did to an extent OTL).
It's honestly a bit of a copout to say that a srrict two party system is inevitable because of FPTP or the EC system. Change a few events slightly (like no Wilson, or a later WWII, or an alternate George Wallace strategy, and butterflies might lead to a different partisan landscape even as the basic sociological tensions remain as OTL.