In order to operate a plasma focus at high axial sheath velocities, the geometry of the inner electrode (anode) of a 3 kJ Mather-type device was modified. Double-stage stepped-anode configurations were tested with the speed-enhanced region sufficiently long 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 μs−1 were achieved. The neutron and soft X-ray productions were found to be sheath-velocity-dependent. A new scaling law for the non-beam component of the neutron yield is proposed.