No, when it comes to unemployment, they already have the records so they will reconcile on their end. Provided they do not accidentally issue an additional refund, you do not have to do anything. If you should receive or be taxed more than owed/due, then they will work it out in next years' return. Given that it is a State issue concern for you, you can go right to your State DOR and rectify it with them. I hope this helps.
