  If M is a closed simple 3-manifold whose fundamental group contains a genus-g
surface group for some g>1, and if the dimension of H_1(M;Z_2) is at least
max(3g-1,6), we show that M contains a closed, incompressible surface of genus
at most g. This improves the main topological result of part I, in which the
the same conclusion was obtained under the stronger hypothesis that the
dimension of H_1(M;Z_2) is at least 4g-1. As an application we show that if M
is a closed orientable hyperbolic 3-manifold with volume at most 3.08, then
H_1(M;Z_2) has dimension at most 5.
