Yes, if you just convey the fact that you overslept without specifying for how much time you overslept then you go with проспать.
However, if you tell the exact time for how much you overslept you go with переспать.
For example, I remember times from my childhood when my mother used to wake me up with these words:
Вставай уже, и так уже 20 минут переспал
Let's say for I should have woken up at 7:00 but I had been still sleeping at 7:20. There we see those 20 minites![]()