Helping America Re-Develop High-Quality Accessible Training Act or the HARD HAT Act This bill prohibits an executive agency from awarding certain contracts unless the contractor requires at least 20% of its non-management employees to participate in a registered apprenticeship program. Specifically, this applies to any contract exceeding $1.5 million for the construction or resurfacing of roads, bridges, or railways; or the construction of subsurface structures, sewage and waste facilities, or public buildings. The agency shall take into consideration the percentage of its non-management employees that complete the program before awarding any subsequent contract.