How to mitigate impact of simultaneous staff breaks in Theme Hospital?

A situation that often happens to me is that I hire a bunch of staff at once in Theme Hospital either at the start of the hospital or during a major expansion, and from that point on, they all go for their staff breaks en masse as their tiredness values are in sync. This devastates the hospital as they leave many rooms vacant, and further staff rush out of their rooms to replace the missing staff, before rushing back to another room after treating just one or two patients, grinding my hospital to a halt.

In some levels I have the spare cash to just keep loads of spare staff hanging around (which causes the game to nag for having staff doing nothing), but in others that is not an option.

How do I prevent or deal with this kind of situation?

I'm afraid I can only offer the most obvious answer: don't hire them all at once. At the start you can hire some before you start building, and some at just before opening (which will help a little).

The primary benefit of not hiring all at once is that it staggers their breaks, but also has a rather large secondary benefit: Better quality staff. The staff hiring pool refreshes every so often, giving you a better selection.

Your low-quality bulk-buy method may also be why you're suffering with breaks so much; I seem to remember lower quality staff get tired more easily.