When dealing with infinities you need to be really precise, and very careful. When you ask about "a ring of infinite circumference" I have to ask - what do you mean by that? The only thing I can think of that's related is a "circle if infinite radius", but the usual interpretation of that is a straight line. Then we're back to an infinitely long hallway.
That's kind of my question. Is there a logically consistent concept of an infinite ring where the "first" room is moved into by the "last" occupant, therefore making the problem dependent on the configuration of the rooms? Or is such a thing inherently finite?
Firstly, the whole point of the hotel story is to talk about addition of cardinals. You are moving off topic and to some extent "bike-shedding" - discussing something other than the real point.
In particular, if there are infinitely many rooms then you must be able to create an embedding of the counting numbers into it. Regardless of whether or not that accounts for all the rooms, you then have an infinite chain, one without a concept of a "last room".
It is possible to have infinitely many cycles, and in those cycles the "last room" is followed by the first room, but there are infinitely many of these cycles, so we can order them in a different way, and we get back again the infinite chain with no "last room".
And this matters. We say that two sets are the same size if we can create a matching between them, and we say they are of different sizes of there is no matching. Be careful. Just because your first attempt at a matching doesn't work, that doesn't mean there isn't one. So in this case of the infinite number of rooms, just because one arrangement seems not to work, that doesn't mean that no arrangement will work.
And indeed, the very definition of the number of rooms being infinite means there is a way to arrange the movements of the existing guests to ensure that there are no "last room" problems.
So - what do you mean?