That won't be necessary. Virtually every hotel in the world deals with back-to-back reservations on a regular basis. It is very common.

There are two basic approaches. In one hotel in which I worked, our Property Management System (PMS) had a "rate change" feature. Using that, we'd check the guest in for the full length of stay and set up the correct rate for each night. The system would automatically change the rate. Doing this created a single receipt, which was preferred by most business travelers.

The PMS at a different hotel didn't have a "rate change" module. There, we had to manually check the guest in and out when the rate changed. We always kept the guest in the same room. This method resulted in a receipt for each reservation. The major downside to this approach was that the guest had to get their room key re-encoded each time. But, that was more of an annoyance than a real problem.

I have no clue how Hyatt operates. But, I wouldn't spend time worrying about it.

This is very true. Hotels will want to keep the guest in the same room because switching rooms is a hassle for everyone involved. Also, Housekeeping services for a checked-out room are usually more extensive, which costs the hotel money.