Just wondering how likely it is that SCs would go back to the applications after bringing several people out for interviews and not finding the right fit. It's the question of a desperate person, I know.
It depends on timing. Most committees realize November - early January is when quite a bit of the hiring takes place. If the failed search aligns with that time frame, a committee may opt to re-run an ad. This can be a smart tactic on their part, since overly selective but strong applicants from the fall cycle might be stuck without a position come January/February and thus reconsider where they applied.
That said, the job cycle is not at all over in January. You'll see some new ads come out through at least mid-March; I've seen some ads hit as late as May 1st. It is improbable these will be R1 or elite SLAC positions, though. Remember that many applicants are not fresh from grad school or a post-doc; those of with tenure-track positions often try to move around for various reasons. If such faculty land a new job in the fall hiring glut, there's fair chance their current institution won't learn of the upcoming vacancy until December or even January; this will then lead to a spring search. Keep applying, but if you really want a position this cycle, I'd recommend applying as widely as reasonable for your goals - above all, try to avoid limiting yourself by location.
Yes they do. This happened for like 3 of the jobs I had offers for, because I had done the "I need to say no for now, but come back to me later" thing. Three places came back to me before they invited new people for interviews. They all filled their positions in the second round of interviewing. All of these places had offered me the position in November or December and conducted their new interviews in January/February/March. So I agree with DocJ — this only works if the first round was in the fall.