A pump system is disclosed for use with a fuel system of an engine. The pump system may have a first pump with a first end, a second end, a reservoir located at the second end, and at least one pump mechanism configured to receive fluid from the reservoir. The pump system may also have a second pump mounted to the first pump at the second end and having at least one pump mechanism configured to discharge fluid into the reservoir of the first pump. The pump system may further have a mechanical input operatively connected to the at least one pump mechanism of each of the first and second pumps.