  (abridged) We continue our study of weakly ionized protostellar discs that
are threaded by a large-scale magnetic field and power a centrifugally driven
wind. It has been argued that in several protostellar systems such a wind
transports a significant fraction of the angular momentum from at least some
part of the disc. We model this case by considering a radially localized disc
model in which the matter is well coupled to the field and the wind is the main
repository of excess angular momentum. We consider stationary solutions in
which magnetic diffusion counters the shearing and advection of the field
lines. In Wardle & K\"onigl we analysed the disc structure in the hydrostatic
approximation and presented disc/wind solutions for the ambipolar diffusivity
regime. In K\"onigl, Salmeron & Wardle (Paper I) we generalized the hydrostatic
analysis to the Hall and Ohm diffusivity domains and identified the parameter
sub-regimes in which viable solutions occur. In this paper we test these
results by deriving numerical solutions (integrated through the sonic critical
surface) of the disc equations in the Hall domain. We confirm the predictions
of the hydrostatic analysis and demonstrate its usefulness for clarifying the
behaviour of the derived solutions. We show that the solutions can be extended
to larger scales (so that they also cross the Alfv\'en critical surface) by
matching the localized disc solutions to global wind solutions of the type
introduced by Blandford & Payne. To facilitate this matching, we construct a
library of wind solutions, which is made available to the community. The
results presented in Wardle & K\"onigl, Paper I and this work form a
comprehensive framework for the study of wind-driving accretion discs in
protostellar and other astrophysical environments. This tool could be useful
for interpreting observations and for guiding numerical simulations of such
systems.
