  A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a
perpendicular Jacobi field making sectional curvature -1 with the geodesic. If
in addition, the sectional curvatures of $M$ lie in the interval
$[-1,-\frac14]$, and $M$ is closed, we show that $M$ is a locally symmetric
space of rank one. This partially extends work by Constantine using completely
different methods. It is also a partial converse to Hamenst\"{a}dt's hyperbolic
rank rigidity result for sectional curvatures $\leq -1$, and complements
well-known results on Euclidean and spherical rank rigidity.
