  In this paper, we classify all commutative weakly distance-regular digraphs
of girth $g$ and one type of arcs under the assumption that
$p_{(1,g-1),(1,g-1)}^{(2,g-2)}\geq k_{1,g-1}-2$. In consequence, we recover
[13, Theorem 1.1] as a special case of our result.
