  Let $\bar{G} = (G, \omega)$ be a vertex-weighted graph, and $\delta$ a
divisor class on $G$. Let $r_{\bar{G}}(\delta)$ denote the combinatorial rank
of $\delta$. Caporaso has introduced the algebraic rank
$r_{\bar{G}}^{\operatorname{alg}}(\delta)$ of $\delta$, by using nodal curves
with dual graph $\bar{G}$. In this paper, when $\bar{G}$ is hyperelliptic or of
genus $3$, we show that $r_{\bar{G}}^{\operatorname{alg}}(\delta) \geq
r_{\bar{G}}(\delta)$ holds, generalizing our previous result. We also show
that, with respect to the specialization map from a non-hyperelliptic curve of
genus $3$ to its reduction graph, any divisor on the graph lifts to a divisor
on the curve of the same rank.
