Hire Just One Act of 2013 - Amends the Internal Revenue Code to allow states, for a 120-day period beginning on the enactment date of this Act, to implement an employment assistance voucher program, in lieu of paying unemployment compensation directly to employees, under which an eligible individual is issued an employment assistance voucher and is hired by a participating employer who receives a subsidy from the state for the wages paid to the employee. Defines an "eligible employee" as an individual who has been unemployed for at least six months, who is eligible for unemployment compensation, and who is likely to exhaust such compensation.

Requires a state program issuing employment assistance vouchers to have in effect measures to recoup subsidies made to an employer if such employer has terminated more employees during the 120-day period than such employer has hired under the program.