I managed to lose two Tilley hats, so I decided to buy something I could better afford to lose and bought a Trekmates Bush Hat. It comes with a headnet (which is nice) stowed in a pocket in the crown. However, I discovered that a 100% plastic headnet, stowed there, defeats the cool properties of the 100% cotton hat, so the net has been evicted, to be packed elsewhere if I ever need it.
Even without such an appendage, hooking the second strap behind your head stops the hat suddenly disappearing ahead of you or going over your eyeS in the event of a sudden strong posterior zephyr.
I only use it in extremis. It at least gives you a chance to grab the hat.
I'm mildly paranoid about hat rigging, as supplied with my current hat, and also with my previous Tilley hats. The chin strap part is a bit of a strangulation risk if, for example, I slip down some slope and the hat takes charge of arresting our joint fall. I always modify the supplied arrangement so that the free ends of the cord are joined under the chin with a spring toggle (not too fierce), so that I can yank the whole thing apart if necessary. Of course, the ends need to be smoothly finished off; no stopper knots!