What you're describing is more than just another submit button.
If you're trying to build a referral form, you must ensure that the Munchkin cookie is excluded from form submission.
I also don't think two buttons, one reading "Submit a new lead" and one "Submit form" makes sense from a UX perspective. If I were to click on the first one, I would assume that the current information on the form is gone, not that my data will be submitted and then the form will be reset for another lead. Why not have the "Submit a new lead" button come up after the form is submitted, i.e. so they can choose to reset the form?