FWIW, since we're talking about hats that aren't your regular cycling cap, I'll plug the Henschel Hiker that I just got. Well stitched from stout tho' doubtless synthetic canvas with sturdy mesh crown for excellent ventilation, and since it's more or less a fedora style, the brim does not flop. I ride in it while dragging the dog through his statutory daily 2 miles of exercise. (Kidding, dog is usually pulling me along.) I rather like the slightly more "formal" style.
And segwaying merrily off of this, I enjoy watching the Sharpe's Rifles series set just before, during, and just after the Peninsular War against Napoleon, and I delight to see the vast variety of men's headgear, ancient holdovers and modern pioneers: Wellington's absurd cocked hat, early cowboy hats and fedoras, shakos, and similars to the weird over-the-top peaked caps worn by Soviet bloc and (nowadays) North Korean officers.
Patrick Moore, who usually wears retro-team -- Legnano! BP! Molteni! -- cycling caps for riding, with 1 cm of the brims bent up to improve sightline. I had to rivet the flipup on my Riv cap to keep it up.