  We continue our study of ladder determinantal rings over a field $\mathsf k$
from the perspective of semidualizing modules. In particular, given a ladder of
variables $Y$, we show that the associated ladder determinantal ring $\mathsf
k[Y]/I_2(Y)$ admits exactly $2^n$ non-isomorphic semidualizing modules where
$n$ is determined from the combinatorics of the ladder $Y$: the number $n$ is
essentially the number of non-Gorenstein factors in a certain decomposition of
$Y$. From this, for each $n$, we show explicitly how to find ladders $Y$ such
that $\mathsf k[Y]/I_2(Y)$ admits exactly $2^n$ non-isomorphic semidualizing
modules. This is in contrast to our previous work, which demonstrates that
large classes of ladders have exactly 2 non-isomorphic semidualizing modules.
