Do I need to register the change of address the moment I sign a new rental contract?

The contract of my current mansion ends on 1st April.
Since it will be a peak moving season, I plan to start renting a new place early around Jan or Feb.

So between Feb-March, I will have two places at the same time for me to move stuffs and have some buffer time.

My question is, do I need to register my change of address in Jan or Feb when I sign a new contract?
Or I just need to register the change of address after I officially moved out from my current place on 1st April?

by Formal-Advisor-7002