I was thinking a reform of the Electoral College, if at all possible. Perhaps a removal of "first-past-the-post", and legislation put into place to allow even small parties to compete nationally (as in, they don't need to get 10% of a state's gubernatorial votes to show up on a ticket or whatever)?
Things like the 17th Amendment were passed due to state pressure and a fear of what might be proposed at a convention. You would probably need something many states adopting resolutions in favor of abolishing the Electoral College, or some early version of the
NPVIC, and lots of states calling for a convention to pass such a thing.
But this is something a bit more fundamental than directly electing Senators. Representation remained the same, and only the process of selection changed hands. A proposal to abolish the Senate would have faced much greater opposition from smaller states, just as an anti-EC proposal would.
A really nasty and corrupt election would probably help, at some point, and I'm not talking things like 1876. I mean something like "Rockefeller buys himself 200 electors in defiance of popular vote."