A refined solution technique is presented for the involute model recently developed. 1,2 This consists of developing the proper relations, in particular, the end conditions, for a body composed of N sub-regions, associated with the finite difference solution scheme. An involute conical frustum problem treated earlier with one sub-region is solved by use of a two-sub-region model. Correlation with a fine network PATCHES-III solution is studied, with particular emphasis on predicted stress discontinuities at the (mathematical) interface of the two sub-regions. Observations regarding the general utility of the new model are also given.