  We construct hairy black hole solutions that merge with the anti-de Sitter
(AdS$_4$) Reissner-Nordstr\"om black hole at the onset of superradiance. These
hairy black holes have, for a given mass and charge, higher entropy than the
corresponding AdS$_4$-Reissner-Nordstr\"om black hole. Therefore, they are
natural candidates for the endpoint of the charged superradiant instability. On
the other hand, hairy black holes never dominate the canonical and
grand-canonical ensembles. The zero-horizon radius of the hairy black holes is
a soliton (i.e. a boson star under a gauge transformation). We construct our
solutions perturbatively, for small mass and charge, so that the properties of
hairy black holes can be used to testify and compare with the endpoint of
initial value simulations. We further discuss the near-horizon scalar
condensation instability which is also present in global
AdS$_4$-Reissner-Nordstr\"om black holes. We highlight the different nature of
the near-horizon and superradiant instabilities and that hairy black holes
ultimately exist because of the non-linear instability of AdS.
