Instant Runoff Voting
Visit this terrific site on various alternative methods of voting and how they work! Mathematically interesting but accessible even to the likes of me. There are links at the site to non-partisan activist organizations that promote alternatives to plurality voting.
I've been converted by my daughter, who is a Green. The Greens' preferred method, IRV, has problems which, ironically, thwart third party candidates but there are other alternatives. Anything has got to be better than plurality voting in a winner-take-all system.
The simplest alternative is Approval Voting (see, e.g. Citizens for Approval Voting. Voters check off the candidates they approve of without ranking them and the candidate with the most votes wins. It can't be too flakey since the Mathematical Association of America (32,000 members), the American Statistical Association and the Institute of Electrical and Electronics Engineers (377,000 members).
We now know what's the matter with Kansas: working class social conservatives are sacrificing their economic interests to buy support for their preferred policies on "lifestyle issues" and cynical plutocrats are happy to throw them a bone to get their own economic agenda through. Given the current system however there is no fix because the center will not hold. Neither party can win without the support of socially conservative working class voters but given the current agendas of both parties, working class conservatives lose whichever party they support. And if a viable third party emerged to represent their interests, we could end up with a three-way deadlock instead of a two-way one.
What can't happen in any case is the emergence of any nuanced political position--as distinct from an incoherent politically motivated compromise.