Bifurcation structures in two-dimensional parameter spaces formed by chaotic attractors alone are still far away from being understood completely. In a series of three papers, we investigate the chaotic domain without periodic inclusions for a map, which is considered by many authors as some kind of one-dimensional canonical form for discontinuous maps. In part I, the basic structures in the chaotic region are explained by the bandcount increment scenario. In part II, the fine self-similar sub-structures nested into the bandcount increment scenario are explained by the bandcount adding and bandcount doubling scenarios, nested into each other ad infinitum. Hereby we fixed in both previous parts one of the parameters to a non-generic value and studied the remaining two-dimensional parameter sub-space. In this part III finally we investigate the structures under variation of this third parameter. Remarkably, this step is most important with respect to practical applications, since it can not be expected, that they operate exactly at this non-generic value.