I don't think a multi-party system is quite the right goal to aim for. In the current system in the US, people with preferences underserved by both D and R don't just lack a respected party, they face a giant barrier to even making the lack visible. Green-leaning? Fuck 'em, what'll they do -- vote Republican? To make your preference unambiguous by an actual vote, under our system, you have to be seriously into the fringe. This creates a kind of activation-energy barrier to coordinating on a change: very many people can feel that, say, smoking pot is OK, without their preference mattering -- without it even affecting their vote. This fraction could even exceed 50%, unless it's
common knowledge that it's over 50%: you don't "throw away" your vote when you expect everyone else to also vote strategically according to preferences which aren't common knowledge.
So a voting system with less of a strategy problem could make common knowledge of real voter preferences easier to arrive at -- the preferences people would honestly vote by, given the chance. You could still have just two major parties, but disciplined by a real option of serious competition if they don't address these no-longer-deniable preferences.
Caveats: 1. AFAIK we still need a clearer understanding of how much strategy people exercise in approval or score voting in practice, once they've gotten used to the new system. And 2. these are just my thoughts as a random citizen trying to see to a way to less large-scale insanity. I'm not a political scientist and I haven't seen one say anything like this about common knowledge and voting systems.