  Einstein-Gauss-Bonnet-gravity (EGB) coupled minimally to a $U(1)$ gauged,
massive scalar field possesses -- for appropriate choices of the $U(1)$ charge
-- black hole solutions that carry charged scalar hair if the frequency of the
harmonic time-dependence of the scalar field is equal to the upper bound on the
superradiant frequency. The existence of these solutions has first been
discussed in \cite{Grandi:2017zgz}. In this paper, we demonstrate that the
critical value of the scalar charge results from the requirement of
non-extremality of the charged black hole solutions and the fact that the
scalar field should not escape to infinity. Moreover, we investigate the hairy
black holes in more detail and demonstrate that the branch of these solutions
joins the branch of the corresponding charged EGB black hole for vanishing
scalar field, but is {\it not} connected to the branch of boson stars in the
limit of vanishing horizon radius. This indicates that it is unlikely that
these black holes appear from the collapse of the corresponding boson stars.
Finally, we prove a No-hair theorem for charged scalar fields with harmonic
time-dependence for static, spherically symmetric, asymptotically flat
electro-vacuum black holes in $d$ space-time dimensions and hence demonstrate
that the GB term is crucial for the existence of the hairy black holes
discussed in this paper.
