Trade Adjustment Assistance Extension Act of 2013 - Amends the Trade Act of 1974 to extend through December 31, 2020: (1) the trade adjustment assistance (TAA) program, and (2) the reemployment trade adjustment assistance (RTAA) program.

Makes funds available through FY2020, and for the period beginning October 1-December 31, 2020 (first quarter of FY2021) for training of adversely affected workers, employment and case management services, and job search expenses and relocation expenses.

Reauthorizes appropriations: (1) through December 31, 2020, for the TAA program for workers; and (2) through FY2020, and for the first quarter of FY2021, for the TAA program for firms and farmers.

Amends the Trade Adjustment Assistance Extension Act of 2011 to declare that TAA program requirements in effect as of February 13, 2011, under the Trade Act of 1974 shall apply to petitions for certification to apply for TAA for workers, firms, and farmers that are filed before January 1, 2021.