My question is what happens next? Do they simply select their next
best candidate? Do they end the hiring process for that opportunity
altogether, or do they restart the hiring process completely?
Every company I have ever worked for would move on and make an offer to the next best candidate, assuming the next best one was still good enough to be hired.
When I hire, I try to narrow the list down to 2-3 candidates for just this reason. I make an offer to the best candidate, and I keep the other 1 or 2 on hold (I don't let them know that they aren't the top choice). If the best declines I move on to the other acceptable candidates on my list.
Other companies might do things differently. Perhaps your contact knows how they handle things at this company.
Based on the fact that I actually know that the opportunity is still
(potentially) open, is there anything I can do to push my case, or do
I just hope they will select the next best candidate and hope that it
will be me?
I don't think there's anything constructive you can do in this case, unless you happen to know that everyone other than the first choice was informed that they were not chosen.
If that were the case, and you suspect that you were a "finalist" you could consider talking with HR and indicating that you are still available. You could also ask your contact if she/he would put in a good word for you.
It couldn't hurt.