My wife has gone through a few. The scarf was fine at lower speeds but had to be held at anything above surface street speed. A friend of hers uses a scarf in their classic Porsche with good success by folding a square in half, draping it over her head, then wrapping back behind her neck and tieing it there.
A ball cap worked pretty well but wasn't good if she wanted to take it off during the day, and it always threatened to fly off at high speed. Remember we drive in all kinds of convertibles, not just MGBs, and some are far more gusty than others!
She found a "cute" English driving cap that she adores and has worn it quite a bit. It stays put and keeps her warm in chilly weather. We never put the top up so this is a plus!
The latest she found is by far her favorite, and very popular in various styles for men and women in the antique cars due to their "in your face" kind of wind. There is a drawstring to hold in on your head in any kind of wind which has proven quite useful. It also allows her to hang it behind her neck if she needs to take it off in the car, without risk of losing it in the wind.
Here is a photo showing the style, which is called a Tilley Hat. Also here it a photo showing that it's very versatile, working in both extreme heat (110+ degrees) and cold.