You could also just have a parliamentary form of government work its way into the United States Constitution as well, but that's less fun.
Edit: Ooh, latest possible. Hmm... Have the AV be implemented as a part of Progressive era's reforms?
Maybe a diffrent Great Depression and no FDR calls for the states to bring about a constitutional convention, and they reorganize the Government in a Parliamentary fashion, or one more open to third parties as a side effect.
How would having a parliamentary government make it more likely for third parties to thrive?
As for AV itself, I'm not so sure that makes it easier for third parties to win. Look at Australia, which I believe is the largest place using AV. The same parties get almost all of the seats. The Liberals, Nats, and Labor all predate the use of AV. The Greens, who have certainly grown with the PR Senate, get the same percentage of votes as the Nationals yet end up with a tenth of the seats.
Parliamentarianism has nothing to do with voting systems. If the U.S. became a parliamentary state tomorrow, the same two parties would win the same percentage of the time. If you want a multiparty state, you need proportional representation in some fashion or large regional parties.
If you want a 19th Century split in the U.S. here is one:
- The Democrats split during or after the Civil War.
- The Solid South votes for the Southern Party. Most of their candidates would have been straight up Democrats OTL.
- The Democratic Party remains in the North.
- The Southern Party expands into a few Western states setting up a handful of Three Cornered Contests. In the gap, a fully independent Progressive Party sets up and wins a few seats and then tries to compete in the East as well.
In such a timeline, I wonder: Would the U.S. adopt some form of PR to balance the interests of the 4 parties? Or would people just accept that party strength at the polls and representation in the House have nothing to do with each other and live with a party with 30% of the votes dominating Congress and the White House?