  In earlier work of NK new closed embedded smooth minimal surfaces in the
round three-sphere $\mathbb{S}^3(1)$ were constructed, each resembling two
parallel copies of the equatorial two-sphere $\mathbb{S}^2_{eq}$ joined by
small catenoidal bridges, with the catenoidal bridges concentrating along two
parallel circles, or the equatorial circle and the poles. In this sequel we
generalize those constructions so that the catenoidal bridges can concentrate
along an arbitrary number of parallel circles, with the further option to
include bridges at the poles. The current constructions follow the Linearized
Doubling (LD) methodology developed before and the LD solutions constructed
here can be modified readily for use to doubling constructions of rotationally
symmetric minimal surfaces with asymmetric sides (work in progress). In
particular they allow us to develop in this forthcoming work doubling
constructions for the catenoid in Euclidean three-space, the critical catenoid
in the unit ball, and the spherical shrinker of the mean curvature flow.
  Our constructions here allow for sequences of minimal surfaces where the
catenoidal bridges tend to be "densely distributed", that is do not miss any
open set of $\mathbb{S}^2_{eq}$ in the limit. This in particular leads to
interesting observations which seem to suggest that it may be impossible to
construct embedded minimal surfaces with isolated singularities by
concentrating infinitely many catenoidal necks at a point.
