Growth in the universal cover under large simplicial volume

Consider a closed manifold M with two Riemannian metrics: One hyperbolic metric, and one other metric g. What hypotheses on g guarantee that for a given radius r, there are balls of radius r in the universal cover of (M,g) with greater-than-hyperbolic volumes? We show that this conclusion holds for all r >= 1 if (Vol(M,g))(2) is less than a small constant times the hyperbolic volume of M. This strengthens a theorem of Sabourau and is partial progress toward a conjecture of Guth.