You should get some of the rent back if another tenant moved in before the 30 days is up. The landlord cannot double rent it--he can't give two different tenants possession at the same time. If you paid for a month's rent, you have possession for that month (for the 30 days you gave notice) even if you choose to move out early--it's still your premises, even if you are not using it. The landlord can only rerent it if he return some of your money to you--the amount for the dates from and after when the new tenant moves in--since if he doesn't return that money, it's still your unit and the new tenant cannot move in.If the landlord will not return some of your rent, you could sue him (e.g. in small claims court) for it.