  In this paper, we study almost finiteness and almost finiteness in measure of
non-free actions. Let $\alpha:G\curvearrowright X$ be a minimal action of a
locally finite-by-virtually $\mathbb{Z}$ group $G$ on the Cantor set $X$. We
prove that under certain assumptions, the action $\alpha$ is almost finite in
measure if and only if $\alpha$ is essentially free. As an application, we
obtain that any minimal topologically free action of a virtually $\mathbb{Z}$
group on an infinite compact metrizable space with the small boundary property
is almost finite. This is the first general result, assuming only topological
freeness, in this direction, and these lead to new results on uniform property
$\Gamma$ and $\mathcal{Z}$-stability for their crossed product $C^*$-algebras.
Some concrete examples of minimal topological free (but non-free) subshifts are
provided.
