  Let $s$ be a point in a polygonal domain $\mathcal{P}$ of $h-1$ holes and $n$
vertices. We consider a quickest visibility query problem. Given a query point
$q$ in $\mathcal{P}$, the goal is to find a shortest path in $\mathcal{P}$ to
move from $s$ to see $q$ as quickly as possible. Previously, Arkin et al. (SoCG
2015) built a data structure of size $O(n^22^{\alpha(n)}\log n)$ that can
answer each query in $O(K\log^2 n)$ time, where $\alpha(n)$ is the inverse
Ackermann function and $K$ is the size of the visibility polygon of $q$ in
$\mathcal{P}$ (and $K$ can be $\Theta(n)$ in the worst case). In this paper, we
present a new data structure of size $O(n\log h + h^2)$ that can answer each
query in $O(h\log h\log n)$ time. Our result improves the previous work when
$h$ is relatively small. In particular, if $h$ is a constant, then our result
even matches the best result for the simple polygon case (i.e., $h=1$), which
is optimal. As a by-product, we also have a new algorithm for a
shortest-path-to-segment query problem. Given a query line segment $\tau$ in
$\mathcal{P}$, the query seeks a shortest path from $s$ to all points of
$\tau$. Previously, Arkin et al. gave a data structure of size
$O(n^22^{\alpha(n)}\log n)$ that can answer each query in $O(\log^2 n)$ time,
and another data structure of size $O(n^3\log n)$ with $O(\log n)$ query time.
We present a data structure of size $O(n)$ with query time $O(h\log
\frac{n}{h})$, which also favors small values of $h$ and is optimal when
$h=O(1)$.
