  In two former papers, the authors independently proved that the space of
hyperbolic cone-3-manifolds with cone angles less than 2{\pi} and fixed
singular locus is locally parametrized by the cone angles. In this sequel, we
investigate the local shape of the deformation space when the singular locus is
no longer fixed, i.e. when the singular vertices can be split. We show that the
different possible splittings correspond to specific pair-of-pants
decompositions of the smooth parts of the links of the singular vertices, and
that under suitable assumptions the corresponding subspace of deformations is
parametrized by the cone angles of the original edges and the lengths of the
new ones.
