  We revisit the phenomenon where, for certain domains $D$, if the squeezing
function $s_D$ extends continuously to a point $p\in \partial{D}$ with value
$1$, then $\partial{D}$ is strongly pseudoconvex around $p$. In $\mathbb{C}^2$,
we present weaker conditions under which the latter conclusion is obtained. In
another direction, we show that there are bounded domains $D\Subset
\mathbb{C}^n$, $n\geq 2$, that admit large $\partial{D}$-open subsets
$\mathscr{O}\subset \partial{D}$ such that $s_D\to 0$ approaching any point in
$\mathscr{O}$. This is impossible for planar domains. We pose a few questions
related to these phenomena. But the core result of this paper identifies a new
family of holomorphic homogeneous regular domains. We show via a family of
examples how abundant domains satisfying the conditions of this result are.
