Strengthening Employment Clusters to Organize Regional Success Act of 2013 or SECTORS Act of 2013 - Amends the Workforce Investment Act of 1998 to require the Secretary of Labor to award renewable three-year competitive industry or sector partnership grants to eligible entities to develop strategies that: (1) encourage growth and competitiveness through work with employers within a targeted industry cluster; (2) help workers move toward economic self-sufficiency and ensure that they have access to supportive services; (3) address the needs of firms with limited human resources or in-house training capacity, including small- and medium-sized firms; and (4) coordinate with entities that carry out state and local workforce investment, economic development, and education activities.