Your new prospective employer will independently apply for a H1B visa. If you have already been counted against the cap, then they can take advantage of that fact in relation to your start date. You can ask your prospective employer what documentation they require from you.
If you have no documentation to show that you have a U.S. position with your current employer under your H1B visa, then you're not going to be successful in having your visa stamped. Your current employer can withdraw its H1B petition if it finds out that you're looking for other jobs or if you resign.

