You quit your job and are on vacation the whole last month of your employment period. What would happen, if you start a new job in this month, besides to earning two wages?
This seems kind of unethical, but from my point of view there is no damage to any involved parties (You, your current employer, your future employer, the government). Is there anything that forbids this practice? Who could find out, that you have two jobs simultaneously and how would they try to punish you?