  We construct a new class of charged, rotating hairy black holes in a
consistent truncation of $\mathcal{N} = 8$ supergravity, which retains one
charged scalar field and a U$(1)$ gauge field. These hairy solutions can be
uplifted to solutions of type IIB supergravity with AdS$_5\times$S$^5$
asymptotics. We find rotating hairy black holes with finite entropy arbitrarily
close to the supersymmetric bound - the resulting supersymmetric solution is a
one-parameter extension of the Gutowski-Reall solution. These solutions have
finite curvature invariants (including at extremality), but in the extremal
limit exhibit diverging tidal forces in the near horizon region. Nevertheless,
we argue that these limiting supersymmetric black holes can be consistently
studied within the supergravity approximation.
