A pilot-operated valve has a main poppet controlling fluid flow between inlet and outlet ports in response to pressure in a control chamber. Fluid flow through a pilot passage, linking the control chamber and the outlet port, moves an exhaust spool between first and second positions to control fluid flow through passages in the main poppet. The first position blocks flow between the control chamber and the outlet port and provides a path between the control chamber and the inlet port. In the second position, flow between the control chamber and the outlet port is enabled, and a second path, that is more restricted than the first path, is provided between the control chamber and the inlet port. When the valve closes, flow through the first path allows movement of the main poppet at a faster rate than during opening when flow occurs through the more restricted second path.