We continue an investigation into the cycle structure of the iterated function system comprising the maps  and  with  and  The perigee of a cyclic orbit is the point nearest the origin. In earlier work, we defined certain equivalence classes of periodic orbits based on their perigee representations, and we exhibited an explicit expression corresponding to the maximal perigee in each class. Here we show how maximal perigees develop across classes as the cycle length increases. This leads in a natural way to the organization of maximal perigees into families and their component branches. We then derive a rational function that attains all maximal perigee values in a given branch.