Flex-Move
Reduce the load the movers will have by hauling whatever boxes will fit into your car to the new digs after the closing and before the move date. Know what rooms you want the furniture to go ahead of time.
Try to plan ahead to reserve space for the mover's van at both destinations. If they have to walk too far to the front door, you're liable for the dreaded "long carry charge" which can add 5% to your bill.
Finally, don't forget the tip (though it isn't expected). It's within reason to give $20 to $30 per mover for a day's work -- more if they've been exceptional, less if you needed them for only four hours.
In some locales, it's customary for the customer to spring for lunch with all-day moves. Also, you should always provide the movers with beverages. They're working up a sweat for you, after all.