The weighted region problem is the problem of finding the weighted shortest path on a plane consisting of polygonal regions with different weights. For the case when the plane is tessellated by squares, we can solve the problem approximately by finding the shortest path on a grid graph defined by placing a vertex at the center of each grid. In this note, we show that the obtained path admits $(\sqrt{2}+1)$-approximation. This improves the previous result of $2\sqrt{2}$.