I've always thought a tenant move was problematic no matter what. How do they leave my unit on the 31st and move into the new unit on the 1st -- they spend the night with a loaded u-haul at a local motel, or they stay with friends, or ...? I have to assume they have the keys to the new place in hand, they load the u-haul, go to dinner and a move and voila, it's midnight, now they can unload into the new place. Can't remember how I did it way back in my wasted youth as a tenant.
Probably the same way I did it. You saved enough money that you could pay the new SD and the rent at the new place. Then you actually rented 2 places for a month, giving yourself plenty of time to move and clean.