If some party other than the Republicans is likely to rise to prominence, I would expect it to be the Libertarian Party. Their ideals seem to be rather similar to the Rs, but with a prettier wrapper that many regular Americans might find palatable, due to the lack of a theocratic bent.
However, the Libertarians seem to be too focused on fuck you, I do what I want to have the discipline to get adequately organized on the scale that a large party needs in order to succeed.
The non-Trumpists could also be the ones to form a Tea-Party-like group, but I doubt there would be enough of them to do what the tea party did.
Our system is so locked into 2 parties, but it seems like this could present one of the best chances for a moderate party* to emerge.
The Democratic base generally wants to move left, and there are strong, popular voices there. Meanwhile, people have been driven out of the Republican party because they can’t abide Trumpism. And the Republican party has pretty much become the party of Trumpism and authoritarianism. I think that either the Republican establishment has to find a way to steer back toward the middle, or they will lose the non-Trumpists.
An emerging moderate party could perhaps pick up both conservative Democrats and the “RINOs” looking for a place to land.
*All references to political continuum are based on US conventions. I know they are all shifted to the right compared to other parts of the world.
When it does rebrand, it tends to happen as a split and a reformation as people choose different sides - and it happens historically quickly - over two or three elections. I feel sorry for Progressive Democrats who just finally felt like they were getting some traction and are going to need to just be single issue anti-facists while Democrats make room in their big tent for pro-Democracy anti-racists Republicans. Because there aren’t many ways to get rid of the nutjobs over there right now, except to starve them out. And splitting big votes just means you can easily elect a nutjob Senator with 30% of the vote because the Greens and the Democratic Socialists and the Neo-Liberals and the NeverTrumpers all put up their own candidates - while the “We are the nutjobs” party stays united.
@Dangerosa: perfectly said. The D’s can sweep the field by absorbing the non-nutjob Rs. But in so doing they’ll end up being all but indistinguishable from, say, the 1960s Republicans but with much better race relations.
The left wing of the Ds needs to be OK with that or else the excess fervor of the nutjob Rs coupled with the the rural and small-state inbuilt electoral advantages will ensure Congress will be 50%+/-2% R nutjob and the senate will be 60%+/-2% R nuthob for the next 20+ years.
And sadly, we’re real close to that mantra repeated against the idea of putting democracy into the Middle East: “One man, one vote, one time”, IOW that whoever won the first free election in e.g. Iraq or Libya would promptly morph into a one party strongman rule and there would never be another genuine democratic vote of the whole people.
If we let the Rs win just once while the nutjobs are more than a tiny fraction, that will be the last free & democratic election the USA holds. High stakes indeed.
And from the look of the House vote on impeachment, the “Republicans fall in line” adage is quite accurate…
I don’t think there is any chance of a right wing wacko party beyond the sort that we currently have. As I see it there are two scenarios. Which occurs depends on how the Proud boys, militias, and Qannon folk react to the reality of a Biden presidency.
If they slink back to their keyboards and lay low, then I think the status quo is likely maintained. The far right faction remains a critical component of Republican primary base, which all elected Republicans have to accommodate or risk losing their jobs.
However, if instead what we saw at the capitol is just the start of a longer campaign of violent acts continuing through the spring and summer as the patriots of freedom resist the illegal and tyrannical Biden administration. then I think the calculus changes. At this point many who may have liked the movie version of revolution, with the good guys winning taking out the bad guys and no one else being hurt may think different when confronted with the reality of innocent blood on the sand. Then there will be strong pressure for the Republican politicians to cut all ties between these extremists, or risk losing even otherwise safe districts.
At that point those who are still bent or Revolution and outright racism, will be where they should be alienated from both sides without a party of their own. This could theoretically led to a splinter party, but given the nature of these people I doubt that they have the organization or patience to actually do the work of finding candidates and getting them on ballots. The whole point of the movement is that the political system is irrevocably corrupt and can only be solved by armed revolution. So going the third part route is a non-starter.
And honestly, better environmental policies. But there isn’t room for a Green Revolution or Medicare for All. There is no Sanders or AOC nomination for President. There are incremental gains. (I’m sort of fond of the 1960s Republican party - without the racism - myself. I’d probably be a Republican if we were pre-Reagan. Though both healthcare and climate change in this country need to be addressed in a more urgent matter).
Oh, and I worry about those Progressive Democrats - they are really fickle and impatient, and too often have some of the same blinders as some of the nuts on the right wing. I just saw a friend of a friend post that Biden stole the nomination from Bernie. 
This is correct and probably the biggest takeaway I got from studying political science in multiple concentrations at a few different universities, other than the fact that I hate politics and have never been employed in anything remotely related. Our system is set up in a way to make it completely impossible to have a viable 3rd party on the national level.