  This paper continues the study of the ellipsoid embedding function of
symplectic Hirzebruch surfaces parametrized by $b \in (0,1)$, the size of the
symplectic blow-up. Cristofaro-Gardiner, et al. (arxiv: 2004.13062) found that
if the embedding function for a Hirzebruch surface has an infinite staircase,
then the function is equal to the volume curve at the accumulation point of the
staircase. Here, we use almost toric fibrations to construct full-fillings at
the accumulation points for an infinite family of recursively defined
irrational $b$-values implying these $b$ are potential staircase values. The
$b$-values are defined via a family of obstructive classes defined in
Magill-McDuff-Weiler (arxiv:2203.06453). There is a correspondence between the
recursive, interwoven structure of the obstructive classes and the sequence of
possible mutations in the almost toric fibrations. This result is used in
Magill-McDuff-Weiler (arxiv:2203.06453) to show that these classes are
exceptional and that these $b$-values do have infinite staircases.
