1938 via The Costume Institute of the Metropolitan Museum of Art
I have only two netted hats in my collection, and one can be worn with or without the netting. I do have the netting to add to some of the other hats, but this is strangely difficult to do, much harder than one would think.
In the "Tarantula" hat. It's made of three prongs of brown fur and a velvet bow, and looks a little like part of a fuzzy spider perched on ones head, with netting. I love it, of course. It's a vintage find from Decades of Fashion on Height Street in San Francisco. They aren't cheap. I think I paid about $40 for this tiny thing.
I do sort of love the idea of a tiny follie of a hat with netting coming off of it. Or even, no hat at all.
Of course it is more common to have a larger follie, one that perches forward with the net coming off of it. Netting shows up on vintage hats of all different shapes and sizes. Here are some 1930s, 1940s, and 1950s examples.
And lastly, one for Prudence to wear . . .
Riding Hat 1905 The Metropolitan Museum of Art