  Let $X$ be a convex cocompact hyperbolic surface and let $\delta$ be the
dimension of its limit set. Let $N_X(\sigma,T)$ be the number of resonances for
$X$ inside the box $[\sigma,\delta]+i[0,T]$. We prove that for all $\sigma >
\delta/2$ we have $$ N_X(\sigma,T) \ll_\epsilon T^{1 + \delta - \frac{3\delta
+2}{2\delta + 2} (2\sigma - \delta) + \epsilon}. $$ This improves upon previous
"improved" fractal Weyl bounds of Naud \cite{Naud14} and Dyatlov \cite{Dya19}.
Moreover, this bound implies that there are resonance-free rectangular boxes of
arbitrary height in the strip $$ \left\{ \delta - \frac{\delta^2}{6\delta+4} <
\mathrm{Re}(s) < \delta \right\}. $$ We combine the approach of Naud
\cite{Naud14} with the refined transfer operator machinery introduced by
Dyatlov--Zworski \cite{DyZw18}, and we use a new bound for certain oscillatory
integrals that arise naturally in our analysis.
