I would absolutely encourage you to submit another offer. Unless the listing agent has an agreement signed that prohibits additional offers they have an obligation to continue to market the house and to submit offers to the Lender. I have had clients who were the 3rd offer submitted get theirs accepted, while another who was the higher bidder (out of 13) be passed over for a lower bid that had a greater down payment. So you never know what will appeal to the Lender. My advice is to make the best offer you are comfortable with, and include the strongest proof of your ability to buy, be it cash or a "pre-qual" letter that is based on an actual review of your credit, your down payment, your employment, etc., and says so! Those details can make a difference to a lender.