  Let $\mathcal{O}_n$ be the set of all maximal outerplanar graphs of order
$n$. Let $ar(\mathcal{O}_n,F)$ denote the maximum positive integer $k$ such
that $T\in \mathcal{O}_n$ has no rainbow subgraph $F$ under a $k$-edge-coloring
of $T$. Denote by $M_k$ a matching of size $k$. In this paper, we prove that
$ar(\mathcal{O}_n,M_k)\le n+4k-9$ for $n\ge3k-3$, which expressively improves
the existing upper bound for $ar(\mathcal{O}_n,M_k)$. We also prove that
$ar(\mathcal{O}_n,M_5)=n+4$ for all $n\ge 15$.
