I used a combination of addons, Magic UV for copy/paste of UV mapping
… and UV Quad Unwrap to get the clean quads without the distortion you sometimes get with “follow active quads”.
I marked the seams for each grid face, then did unwrap, which gives you rounded faces, I then selected one face and select linked with delimit by seam, ran the UV Quad unwrap. selected those faces, copied the UV, inverted the selection, pasted the UV, then in the UV window just hit CTRL+P to pack.
To be honest, given the curvature, you really don’t want purely straight quads as the texture will distort slightly when applied, given the surface is actually curved. But htat said, it is good for adding straight lines on a curved surface.