Under mild topological restrictions, we obtain new linear upper bounds for the dimension of the mod pp homology (for any prime pp) of a finite-volume orientable hyperbolic 33-manifold MM in terms of its volume. A surprising feature of the arguments in the paper is that they require an application of the Four Color Theorem. If MM is closed, and either (a) π1(M)π1(M) has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus 22, 33 or 44, or (b) p=2p=2, and MM contains no (embedded, two-sided) incompressible surface of genus 22, 33 or 44, then dimH1(M;Fp)<157.763⋅vol(M)dimH1(M;Fp)<157.763⋅vol(M). If MM has one or more cusps, we get a very similar bound assuming that π1(M)π1(M) has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus gg for g=2,…,8g=2,…,8. These results should be compared with those of our previous paper “The ratio of homology rank to hyperbolic volume, I,” in which we obtained a bound with a coefficient in the range of 168168 instead of 158158, without a restriction on surface subgroups or incompressible surfaces. In a future paper, using a much more involved argument, we expect to obtain bounds close to those given by this paper without such a restriction. The arguments also give new linear upper bounds (with constant terms) for the rank of π1(M)π1(M) in terms of volMvolM, assuming that either π1(M)π1(M) is 99-free, or MM is closed and π1(M)π1(M) is 55-free.