A flow rate control system includes a housing and a valve assembly slidingly disposed within a housing inner bore. The housing includes bypass openings. The valve assembly includes a valve and an orifice disposed in a valve inner bore. The valve includes a plurality of valve bypass bores extending axially through a valve collar. The valve assembly slides between closed and fully open positions. A spring biases the valve assembly toward the closed position in which the valve closes the housing bypass openings. In the open position, a bypass fluid path is formed including the valve bypass bores and the housing bypass openings. The valve assembly is flow rate controlled in the closed position and pressure controlled in the open position. The valve assembly may slide within a sleeve assembly including sleeve bypass openings, which connect the valve bypass bores and housing bypass openings in the bypass fluid path.