You don't need a new 656 Form if you write within a month. Instead, write a letter if the new Offer is not radically different from your old one and if your financial circumstances have not appreciably changed. To submit a significantly different Offer, complete another Form 656. The Revenue Officer or Special Procedures Officer might help you come up with a way to make your Offer acceptable. Looking at Offers in Compromise that were accepted might give you an idea of what kinds of proposals the IRS goes for.
Some Districts accept far more Offers than others.
Go to your local IRS office and ask to see all Offers that were accepted within the last year. It's a good idea to call the IRS first and tell them you are coming. Accepted Offers are a matter of public record, for one year, although these Offers won't have any direct bearing on yours. If your Offer in Compromise is rejected, you will receive a letter from the IRS with an explanation. You can either call the person who signed the letter and try to get her to change her mind, or you can formally appeal.