To study the effect of higher axial sheath velocities on neutron production, the geometry of the inner electrode (anode) of a 3-kJ Mather-type plasma focus device was modified. Double-stage stepped-anode configurations were tested with a speed-enhanced region long enough for a significant increase in speed but not long enough to allow the development of force-field flow-field separation at the end of the axial phase. Peak axial speeds up to 15 cm/s were achieved without modifying the deuterium filling pressure and charging voltage. The neutron and soft X-ray productions were found to be dependent on sheath velocity. A new scaling law for the nonbeam component of the neutron yield was proposed.