Those wideish brimmed Auzzie outback, bush type hats, similar to what Highlander suggests are Ace. They work all year round and I have found them the only thing to keep your head dry 360 degrees and the rain out of your face at the same time. You can work outdoors in them in a bit of rain and not even notice it is raining. Without it then rainwater would be streaming down your face and into your eyes. they are excellent for avoiding this and whilst keeping the head dry, they do not obstruct the hearing - unlike hoods or earflap type hats. It is actually possible that they help to redirect some sound more towards the ear and so to slightly amplify the sound - dependent on direction in relation to the brim. The earlobe effect thing going on! I would advise a brim large enough to deflect most water away from the neck/clavicle but not too wide as the brim can be a hindrance if it flops about in a wind. As you point out SD, the only downside to these style hats is stability in a wind or whilst running. The tried & tested way to compensate for this is to use a chin strap when needed or to let it drop behind over your back on its strap... Simples.. :-)
Alternatively you could always try one of those tennis/golf type forehead peak on an elastic head band.. lol
What about one of those under the brimmed hat, so that if you have to slip it sown behind your back, then the peak will still be there??
Also the wide brimmed hat may act as an air brake if you were leaping from buildings etc.. lol TL.
A serious note - I have tried one of those fold up auzzie hats and it was a disappointment to the point of being dangerous! The problem is that it is not waterproof and at the same time is not of a heat retaining material. I found that even after the copious application of wax, the rain would soak through to the skin. Whilst I did have the benefit of no water running down my face, I also received a very cold head and the temp was still dropping until I removed it. In our temperate climate with cold periods, then the hat is actually a medical liability. I advise all readers out there to take this consideration on board when looking at any hat choice. I suspect it applies to non collapsible/foldy versions and any other type of hat as well. In addition to this, some of the very nice surplus winter gear out there is actually from countries that do not get much rainfall at all even though they are very cold. This results in their particular design of gear being ideal for their national climate, but not suited to ours, no matter how snug it may look. If you want kit to cover many possibilities, then make a wise informed choice.
(Quality trilbys of yesteryear were made out of fur felt. Now that is a superb outdoor material that retains its heat properties in all weathers.)
"As you point out S13" it should in the text