The Two-Party System in the US only works if you have two big-tent parties that are able to compete all over the place and have a shot at winning in enough places consistently. You can have two parties at the presidential level without this being the case at the legislative level.
In an increasingly partisan environment, that creates lots of opportunities for dissent by outsiders. This last cycle in 2018 a Libertarian candidate for state house in Wyoming came within 53 votes of ousting the Majority Leader, for example. Alaska in 2016 amount to a Libertarian vs Republican Senate race (the Democrat came fourth). The Greens almost took CA's Mayor Office in 2003.
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1. The GOP never absorbs various socially conservative to populist elements from the Democrats and the AIP manages to continue as something of a Southern Parti Quebecois with some popularity among working class whites outside of the region and law-and-order types.
2. Ross Perot manages to translate his 1992 success into a successful 1993 Senate bid to replace Lloyd Bentsen. Frustrated, he forms the Reform Party two years sooner and some folks get elected in 1994 and 1996 (funded by him, of course). Angus King, Lowell Weicker, Bob Smith, Virgil Goode, and other OTL independents and party switchers proceed to join up. Jerry Brown is elected Mayor of Oakland as a Reformer. Matt Gonzalez gets elected Mayor of San Francisco as a Reformer. Dick Lamm and Ed Zschau get elected as Reformers. Reform ends up as a mix of Paleoconservatives, Economic Centrists, Protectionists, Pro-infrastructure, pro-single payer, some members of the antiglobalist left (hence the Green element), and even some members of the black community like Jesse Jackson (who Perot worked with on some stuff, interestingly).
3. Realizing that the GOP has redistricted out of the job, Dennis Kucinich decides to primary Obama in 2012. There was some interest in Obama getting a primary challenger in 2012 and odds are Kucinich would win a few states (West Virginia, Arkansas, maybe Georgia if Kemp keeps Obama off the ballot). Obama ultimately shuts him down, but this translates into a wave of progressive-independents in the house in 2014 (Marianne Williamson, Dennis Kucinich, Cynthia McKinney, Seth Moulton, etc). Also Larry Pressler and Greg Orman get elected as independents, which while a separate matter is still a boost to a "year of the independent" story.
4. Doug Hoffman wins the 2009 New York House race. Jack Davis wins the 2010 New York race in the 26th district as a conservative after it comes out that the GOP candidate sought the services of a crossdressing prostitute. Lisa Murkowski wins the GOP primary, but Joe Miller wins as an Independent or Libertarian. Tom Tancredo wins the Colorado gubernatorial (making the GOP a third party in CO) and Virgil Goode gets reelected to the house (and proceeds to swap parties from R to C). This all inspires a bunch of other conservatives to run outside of the GOP for congress in very red areas.
5. The Minnesota Independence Party manages to retain the governorship in 2002 with Tim Penny. The Independence Party does well, gets some members elected to the house of representatives, a Senator (Barkeley?), and becomes a permanent fixture of state politics. It is a third-party only insomuch as it's a fixture of Minnesota politics that refuses to go away. Eventually it grows to have presences in the Dakotas, Iowa, Wyoming, Montana, Indiana, Nebraska, and Wisconsin as well. It is very much NOT a national party and has no aspirations to be (it's officeholders routinely endorsing democrats and republicans for office in other states and for president), but is seen as a regional oddity that nobody's really bothered by. They never bother to run candidates for president, being more interested in state governance and legislative activity.