  We revisit our earlier work which lead to a periodic table of
Borcherds-Kac-Moody algebras that appeared in the context of the refined
generating function of quarter-BPS (dyons) in $\mathcal{N}=4$ supersymmetric
four-dimensional string theory. We make new additions to the periodic table by
making use of connections with generalized Mathieu moonshine as well as umbral
moonshine. We show the modularity of some Siegel modular forms that appear in
umbral moonshine associated with Niemeier lattices constructed from A-type root
systems and further show that the same Siegel modular forms appear for
generalized Mathieu moonshine in some cases. We argue for the existence of a
new kind of BKM Lie superalgebras that arise from the dyon generating functions
for the $\mathbb{Z}_5$ and $\mathbb{Z}_6$ CHL orbifolds.
