  In the first part of this series, an augmented PDE system was introduced in
order to couple two nonlinear hyperbolic equations together. This formulation
allowed the authors, based on Dafermos's self-similar viscosity method, to
establish the existence of self-similar solutions to the coupled Riemann
problem. We continue here this analysis in the {restricted} case of
one-dimensional scalar equations and investigate the internal structure of the
interface in order to derive a selection criterion associated with the
underlying regularization mechanism and, in turn, to characterize the
nonconservative interface layer. In addition, we identify a new criterion that
selects double-waved solutions that are also continuous at the interface. We
conclude by providing some evidence that such solutions can be non-unique when
dealing with non-convex flux-functions.
