If you won your appeal, I presume you are getting all the unemployment you were entitled to. You can't recover twice by suing you former employer. The above is for informational purposes only and not meant as legal advice.
My colleague's answer is correct: You cannot recover twice for the same injury. However, if it turns out that the employer is reporting factually erroneous information to others and the misstatements prevent you from obtaining a position with a particular employer, you might have a cause of action for defamation, interference with prospective business advantage or "prima facie" tort. Before proceeding down that route, you need to know that those are very difficult claims to prevail upon and very expensive to litigate.
You should consult with an employment law specialist in New York and consider retaining an attorney to draft an appropriate letter to the employer, demanding that it delete any inaccurate information from its files and agree to the manner in which it will respond to future prospective employers.
NOTE: As AVVO's Rules make clear, this answer does not establish an attorney-client relationship nor constitute legal advice. It is meant for general informational purposes only.