  Let $N$ be a compact, orientable hyperbolic 3-manifold whose boundary is a
connected totally geodesic surface of genus $2$. If $N$ has Heegaard genus at
least $5$, then its volume is greater than $2V_{\rm oct}$, where $V_{\rm
oct}=3.66\ldots$ denotes the volume of a regular ideal hyperbolic octahedron in
$\mathbb{H}^3$. This improves the lower bound given in our earlier paper
``Volume and topology of bounded and closed hyperbolic $3$-manifolds.'' One
ingredient in the improved bound is that in a crucial case, instead of using a
single ``muffin'' in $N$ in the sense of Kojima and Miyamoto, we use two
disjoint muffins. By combining the result about manifolds with geodesic
boundary with the $\log(2k-1)$ theorem and results due to Agol-Culler-Shalen
and Shalen-Wagreich, we show that if $M$ is a closed, orientable hyperbolic
$3$-manifold with $\mathop{\rm vol} M\le V_{\rm oct}/2$, then $\dim
H_1(M;\mathbb{F}_2)\le4$. We also provide new lower bounds for the volumes of
closed hyperbolic $3$-manifolds whose cohomology ring over $\mathbb{F}_2$
satisfies certain restrictions; these improve results that were proved in
``Volume and topology$\ldots$.''
