The most obvious avenue is to promote ballot propositions to change the voting mechanism, in states that allow this, like California. California recently change

The most obvious avenue is to promote ballot propositions to change the voting mechanism, in states that allow this, like California. California recently changed to "open primaries", for example. (A bad idea, that one, but approval voting or score voting could go through the same way.) It'd need to get pushed from outside the major parties, I guess by someone genuinely public-spirited, since the benefit is very diffuse and long-term. Unless some Machiavelli can figure out an angle -- I sure can't. Once it's had a toehold in a couple states for a few election cycles, though, I think there's a good chance it'd spread, because it seems hard to argue against once it's on the table as a serious option. (Maybe the strongest move then for the parties would be to muddy the waters with instant runoff voting and other such complications.) (I think approval voting is more 'American' than proportional representation and haven't really thought about how we'd move to proportional rep, but there's the same first step available.)