Can an immigrant stay in the U.S. while H1B visa is considered for the next year due to a full quota

As always, I advise speaking with an attorney directly -especially in situations like this, because I'm not 100% sure I understand the facts as you're presenting them and I don't have the ability to just ask you to explain wha tI might not understand to be sure I'm answering your questions correctly. So, I'll explain what I'm assuming to be true based on your question and then answer based on that - if my assumptions are incorrect, you can't rely on this answer.

I gather that you got a cap-subject H-1B - an H-1B that WAS counted under the quota - and then worked for a little more than five and a half years on this H-1B.

Then, I assume that after using a little more than the five and a half years of the six years allowed on an H-1B, you changed status to F-1 to attend graduate school for an MBA. You would therefore be eligible for one year of work authorization in your field on Optional Practical Training after your graduate, and want to use this up before using more of your H-1B time

As I understand your question, you want to know - if you do this - can you just switch back to the H-1B whenever the Optional Practical Training runs out or can the H-1B only begin only at certain times of the year, as new cap-subject H-1Bs do in years when the cap has been reached for the prior year.

If this is the case, you're fine - you can just switch back to an H-1B for the remaining 14 months at any point. You don't need to wait for the next October 1, since your H-1B has already been counted under the cap. It doesn't depend on the cap OR your employer.

You don't mention anything about permanent residence in your question - whether you want it, or whether you had ever begun a process. If you do want to begin a permanent residence process, be aware of the timing. If you start the process with some form of filing before the sixth year of your H-1B starts, you preserve the ability to extend the H-1B beyond the sixth year if the process doesn't complete in time. You may even be able to extend the end of your sixth year by "recapturing" H-1B time spent abroad earlier. This is something that you should discuss early with an attorney (and with future employers).