National Discovery Trails Act of 2013 - Amends the National Trails System Act to establish national discovery trails which shall be extended, continuous interstate trails located so as to provide for outdoor recreation and travel and to connect representative examples of America's trails and communities.

Authorizes the designation of such trails on federal lands and, with the consent of the owner, on non-federal lands.

Designates as a national discovery trail the 6,000-mile American Discovery Trail which shall extend from Cape Henlopen State Park in Delaware to Point Reyes National Seashore in California, splitting into northern and southern routes at Cincinnati, Ohio, and rejoining at Denver, Colorado.

Sets forth requirements for designation of a trail as a national discovery trail.

Requires the appropriate Secretary for each national discovery trail to: (1) administer the trail in cooperation with a competent trailwide volunteer-based organization; and (2) submit a comprehensive plan for the protection, management, development, and use of such trail.