For the purpose of complying with your new apartment's landlord, it's best to change the address on your policy before you move in. This will satisfy the insurance requirement for your new place. That does not me that your old apartment is completely uncovered. Since a renters insurance
policy provides coverage for your liability, even if you are away from home, you should still be covered. It's best to let your insurance company know about the specifics of your move just to ensure you aren't leaving yourself exposed, but moving before the end of a lease is quite common and should pose an issue for you or your company.