  Recently the first example of a family of pro-$p$ groups, for $p$ a prime,
with full normal Hausdorff spectrum was constructed. In this paper we further
investigate this family by computing their finitely generated Hausdorff
spectrum with respect to each of the five standard filtration series: the
$p$-power series, the iterated $p$-power series, the lower $p$-series, the
Frattini series and the dimension subgroup series. Here the finitely generated
Hausdorff spectra of these groups consist of infinitely many rational numbers,
and their computation requires a rather technical approach. This result also
gives further evidence to the non-existence of a finitely generated pro-$p$
group with uncountable finitely generated Hausdorff spectrum.
