Emergency Unemployment Compensation Extension Act of 2013

Emergency Unemployment Compensation Extension Act of 2013 - Amends the Supplemental Appropriations Act, 2008 to extend emergency unemployment compensation (EUC) payments for eligible individuals to weeks of employment ending on or before January 1, 2015.

Amends the Assistance for Unemployed Workers and Struggling Families Act to extend until December 31, 2014, requirements that federal payments to states cover 100% of EUC.

Amends the Unemployment Compensation Extension Act of 2008 to exempt weeks of unemployment between enactment of this Act and June 30, 2015, from the prohibition in the Federal-State Extended Unemployment Compensation Act of 1970 (FSEUCA of 1970) against federal matching payments to a state for the first week in an individual's eligibility period for which extended compensation or sharable regular compensation is paid if the state law provides for payment of regular compensation to an individual for his or her first week of otherwise compensable unemployment. (Thus allows temporary federal matching for the first week of extended benefits for states with no waiting period.)

Amends the FSEUCA of 1970 to postpone similarly from December 31, 2013, to December 31, 2014, termination of the period during which a state may determine its "on" and "off" indicators according to specified temporary substitutions in its formula.

Amends the Supplemental Appropriations Act, 2008 to appropriate funds out of the employment security administration account through FY2015 to assist states in providing reemployment and eligibility assessment activities.

Amends the Railroad Unemployment Insurance Act to extend through December 31, 2014, the temporary increase in extended unemployment benefits.