Let s be a point in a polygonal domain $${\mathcal {P}}$$ P of $$h-1$$ h - 1 holes and n vertices. We consider a quickest visibility query problem. Given a query point q in $${\mathcal {P}}$$ P , the goal is to find a shortest path in $${\mathcal {P}}$$ P to move from s to seeq as quickly as possible. Previously, Arkin et al. (SoCG 2015) built a data structure of size $$O(n^22^{\alpha (n)}\log n)$$ O ( n 2 2 α ( n ) log n ) that can answer each query in $$O(K\log ^2 n)$$ O ( K log 2 n ) time, where $$\alpha (n)$$ α ( n ) is the inverse Ackermann function and K is the size of the visibility polygon of q in $${\mathcal {P}}$$ P (and K can be $$\varTheta (n)$$ Θ ( n ) in the worst case). In this paper, we present a new data structure of size $$O(n\log h + h^2)$$ O ( n log h + h 2 ) that can answer each query in $$O(h\log h\log n)$$ 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., $$h=1$$ 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 $$\tau $$ τ in $${\mathcal {P}}$$ P , the query seeks a shortest path from s to all points of $$\tau $$ τ . Previously, Arkin et al. gave a data structure of size $$O(n^22^{\alpha (n)}\log n)$$ O ( n 2 2 α ( n ) log n ) that can answer each query in $$O(\log ^2 n)$$ O ( log 2 n ) time, and another data structure of size $$O(n^3\log n)$$ O ( n 3 log n ) with $$O(\log n)$$ O ( log n ) query time. We present a data structure of size O(n) with query time $$O\big (h\log \frac{n}{h}\big )$$ O ( h log n h ) , which also favors small values of h and is optimal when $$h=O(1)$$ h = O ( 1 ) .