Teacher Tax Deduction Enhancement Act of 2013 - Amends the Internal Revenue Code to: (1) extend until 2020 the tax deduction for certain expenses of elementary and secondary school educators (i.e., teachers, instructors, counselors, principals, or aides); (2) increase to $500 the allowable amount of such deduction for full-time educators (working at least 900 hours during the school year); and (3) allow such tax deduction for educators in preschool programs.