Technically both of these approaches are not as good as a wide brimmed hat in terms of face protection. However, for some sports it's superior (anything with wind or clearance issues). I'm on a Stetson at the moment after a US trip but usually go for whatever broad brim is available. Buying new ones gets old so going for a 'crushable' model is the go - I like the felt Stetsons. The other benefit from a broad brim is it functions as an auto-umbrella in light weather which is damn cool. Also, hat-on-face is similar to those plane-sleep-eyewear bands.