  In this paper, we explore different possible choices of expanded
degenerations and define appropriate stability conditions in order to construct
good degenerations of Hilbert schemes of points over semistable degenerations
of surfaces, given as proper Deligne-Mumford stacks. These stacks provide
explicit examples of constructions arising from the work of Maulik and
Ranganathan. This paper builds upon and generalises previous work in which we
constructed a special example of such a stack. We also explain how these
methods apply to constructing minimal models of type III degenerations of
hyperk\"ahler varieties, namely Hilbert schemes of points on K3 surfaces.
