  Let $S$ be a smooth affine surface of logarithmic Kodaira dimension one and
let $(V,D)$ be a pair of a smooth projective surface $V$ and a simple normal
crossing divisor $D$ on $V$ such that $V \setminus \operatorname{Supp} D = S$.
In this paper, we consider the logarithmic multicanonical system $|m(K_V +
D)|$. We prove that, for any $m \geq 8$, $|m(K_V+D)|$ gives an
$\mathbb{P}^1$-fibration form $V$ onto a smooth projective curve.
