You said it...rent from you new buyer. Best option is to waive the contingency and make sure your lender is working with you through and through. Tell your buyer you may need to stay 2-4 days.
The only other option is furniture in storage while you stay in a hotel for a few days....
Probably too late now but in my market - which is very good for sellers and challenging to buy because of limited inventory - sellers are asking for and getting 30 to 60 day lease-backs. That way they have their funds from their sale in hand and time to pin down their purchase.
How to Win a Bidding War on Chicago Real Estate: