  We present exact radially self-similar solutions of special-relativistic
magnetohydrodynamics representing ``hot'' super-Alfvenic outflows from strongly
magnetized, rotating compact objects. We argue that such outflows can plausibly
arise in gamma-ray burst (GRB) sources and demonstrate that, just as in the
case of the trans-Alfvenic flows considered in the companion paper, they can
attain Lorentz factors that correspond to a rough equipartition between the
Poynting and kinetic-energy fluxes and become cylindrically collimated on
scales compatible with GRB observations. As in the trans-Alfvenic case, the
initial acceleration is thermal, but, in contrast to the solutions presented in
Paper I, part of the enthalpy flux is transformed into Poynting flux during
this phase. The subsequent, magnetically dominated acceleration can be
significantly less rapid than in trans-Alfvenic flows.
