What are the rules for travel when extending a residence permit?

Good question. And I am not sure of the answer. My guess would be that you could indeed count the time between as being a visa-free tourist. But you will have to be dealing with German officials about your new job and residency anyway, so I would just ask them. One would expect they have dealt with such a situation often enough and there is nothing unusual or wrong in what you intend to do, so I would not worry too much.

If your residence permit expires and you did not get any other visa or permit (yet), you are an illegal immigrant - which should be avoided at all cost.
Thus contact your nearest Ausländerbehörde now! If you can show the employment contract starting in May, they are usually quite accommodating and will give you a temporary visa to cover the gap - I think most likely a Schengen visa or similar.

beppi wrote:

If your residence permit expires and you did not get any other visa or permit (yet), you are an illegal immigrant - which should be avoided at all cost.
Thus contact your nearest Ausländerbehörde now! If you can show the employment contract starting in May, they are usually quite accommodating and will give you a temporary visa to cover the gap - I think most likely a Schengen visa or similar.

What I took into account is that US citizens don't need or get a Schengen visa for tourism; they get 3 months visa-free. One should not always assume non-EU citizens need such a visa as not only US persons but also Canadians, Australians, New Zealanders and some others are exempt.