Extend Not Cut SNAP Benefits Act - Amends the American Recovery and Reinvestment Act of 2009 to extend the benefit calculation of 113.6 % of the June 2008 value of the thrifty food plan for the supplemental nutrition assistance program (SNAP, formerly known as food stamps) and the consolidated block grants for Puerto Rico and American Samoa until September 30, 2016, or the first date on which the value of the thrifty food plan equals or exceeds 113.6 % of the June 2008 value.