I think it depends on whether the employer wants to be a nice guy.  There is a procedure for refunding the employee and correcting the amount withheld (to zero) before the end of the year.  It doesn't hurt to ask:

(iii) Overcollection discovered in subsequent calendar quarter. Where more than the correct amount of New York State personal income tax has been deducted from an employee's wages and the employer discovers the overcollection after the filing of the quarterly combined withholding and wage reporting return for the calendar quarter in which the overcollection occurred, but prior to filing such employer's last quarterly combined withholding and wage reporting return for the same calendar year, such employer must correct the overpayment to the Department of Taxation and Finance by entering the correct amount of the withholding tax liability on an amended quarterly combined withholding and wage reporting return. The employer must attach to such amended quarterly combined return an explanation of the computation of the overpayment and must also indicate whether or not such employer has repaid the overcollection to the employee. Where an employer shows an overpayment described in this subparagraph on the amended quarterly combined withholding and wage reporting return and has repaid the overcollection to the employee, such overpayment may only be refunded; the employer may not elect to credit the overpayment to a subsequent calendar quarter.

https://govt.westlaw.com/nycrr/Document/I50e94685cd1711dda432a117e6e0f345