  We show that both $\mathfrak{ap=c}$ and $\diamondsuit(\mathfrak{b})$ imply
the existence of MAD families with pseudocompact Vietoris hyperspace,
substantially expanding the list of models where their existence is known. We
also discuss some properties on the structure of fin-intersecting MAD families
by providing some new examples of non-fin-intersecting MAD families with
special properties. We show that the Baire number of $\omega^*$ is strictly
larger than $\mathfrak c$ if and only if for every MAD family $\mathcal A$,
$\Psi(\mathcal A)$ and its Vietoris hyperspace are $p$-pseudocompact for some
free ultrafilter $p$.
