  We present a numerical study of fully non-linear, rotating and charged hairy
black hole solutions in five-dimensional anti-de Sitter space, which originate
from a consistent truncation of $\mathcal{N}=8$ supergravity, and can be
consistently embedded in type IIB supergravity with AdS$_5\times$S$^5$
asymptotics. The hairy black holes have one scalar field charged under a $U(1)$
gauge field, and branch from the near-extremal Cveti\v{c}, L\"{u} and Pope
solutions. We give numerical evidence that the hairy solutions exist
arbitrarily close to the BPS bound for all charges, and saturate it in the
$T\rightarrow 0$ and $T\rightarrow\infty$ limits. We give further evidence for
the conjecture of Markevi\v{c}i\={u}t\.e and Santos, that on the BPS bound, the
rotating hairy black holes form a two-parameter family of solutions with finite
entropy, and can be regarded as a one-parameter extension of the supersymmetric
Gutowski and Reall black hole. We analyse the approach to the supersymmetric
limit and explore the full phase diagram. In the planar horizon limit we find a
two parameter family of rotating hairy black brane solutions which cannot be
obtained via a Lorentz boost. The field theory dual exhibits a spontaneously
generated current. The results of this paper suggest rich and intricate
structure of hairy black hole solutions in AdS$_5\times$S$^5$, and highlight
their importance in understanding the thermodynamics of $\mathcal{N}=4$ SYM.
