Economic Security Act of 2003 - Provides for a program of temporary enhanced unemployment compensation which may add the greater of 15 percent or $25 to the amount of weekly regular compensation (including dependents' allowances) for which an individual is eligible under the Federal-State Extended Unemployment Compensation Act of 1970. Modifies eligibility requirements for regular compensation, where applicable for purposes of benefits under this Act, to: (1) set an alternative base period ending at the close of the most recently completed calendar quarter; and (2) allow individuals to seek less than full-time work or not accept full-time work.Sets forth program requirements for Federal-State agreements, formulas for determining weekly benefits, nonreduction and coordination rules, payments to States, and financing.Applies program agreements to weeks of unemployment: (1) beginning after the date on which such an agreement is entered into; and (2) ending before July 1, 2004. Makes a modification relating to alternative base periods applicable only to initial claims filed after September 11, 2001. Makes a modification relating to part-time employment and increased benefits applicable to weeks of unemployment in such agreement period, regardless of the date on which an individual's claim for benefits is filed.