I think you would have a system where gridlock is generally less likely. If a President had a minority in Congress, but a majority if you add the third party, then suddenly legislation becomes easier to pass, as if the other main party said no to there proposals, then the President could just go to the other party to get the votes they need. There would be competition between parties to work with each other, leaving them more open to compromise. So you would probably see President's able to get more stuff passed congress, and in less watered down forms.
That could be a particularly strong selling point for a party within one region that could keep them going as a viable third party, by holding the balance of seats in Congress, they bring concessions that benefit that region, leaving a region which may have otherwise have been neglected to be the recipient of a disproportionate amount of legislative attention.
Personally I think it would make the whole system work far more smoothly. A system based on compromise is always going to be problematic whilst one party can be pretty confident of scuppering the others proposals completely by voting against them, it gives them too much power. Competition to cooperate with one another would do the US some good. That is why I think it needs some form of PR even more urgently than we do here in the UK.