  We systematically study various sub-leading structures in the superconformal
index of ${\cal N}=4$ supersymmetric Yang-Mills theory with SU($N$) gauge
group. We concentrate in the superconformal index description as a matrix model
of elliptic gamma functions and in the Bethe-Ansatz presentation. Our
saddle-point approximation goes beyond the Cardy-like limit and we uncover
various saddles governed by a matrix model corresponding to SU($N$)
Chern-Simons theory. The dominant saddle, however, leads to perfect agreement
with the Bethe-Ansatz approach. We also determine the logarithmic correction to
the superconformal index to be $\log N$, finding precise agreement between the
saddle-point and Bethe-Ansatz approaches in their respective approximations. We
generalize the two approaches to cover a large class of 4d ${\cal N}=1$
superconformal theories. We find that also in this case both approximations
agree all the way down to a universal contribution of the form $\log N$. The
universality of this last result constitutes a robust signature of this
ultraviolet description of asymptotically AdS$_5$ black holes and could be
tested by low-energy IIB supergravity.
