J. Maher has proven that a closed, connected and orientable hyperbolic
3-manifold $M$ virtually fibers over the circle if and only if it admits an
infinite family of finite covers with bounded Heegaard genus. Building on
Maher's proof, we show in this article that if the genus in a family of finite
covers grows at most sub-logarithmically with the covering degree, then the
manifold $M$ is virtually fibered. We introduce sub-logarithmic versions of
Lackenby's infimal Heegaard gradients. In this setting, we prove the analogues
of Lackenby's Heegaard gradient and strong Heegaard gradient conjectures.