I use lightweight, ventilated, SPF fabric ball caps made by Outdoors Research (OR). This is the same brand as the tan hat pictured two posts up from here. They fold flat (brim folds into the rear of the cap), and can easily go in a rucksack side pocket or a rear side pants pocket. I also wear a whole brimmed "sun hat" similar to the two above. The sun hat also has an adjustable chin-string for windy days.
My reason for having two hats is simple. The ball cap is better for use "in town" or while traveling. It is also better worn under the hood of your rain gear to keep rain off your eyeglasses, to give shape to the brim of your poncho or parka hood, and to preclude rain from channeling around the full brim of a sun hat down your back...
The sun hat, with the brim running all around is best for all sunny or overcast days to keep your head and neck in shade, and as has been described above.
You will wear both types of hats, on different types of days, in different types of weather.
If you are walking during the hot, summer months, where less rain is expected, the single "bucket-style" hat may suffice. But, as I usually walk in late April through early June typically, I can experience anything from snow to bright hot, dry weather along different parts of the
Camino Frances. The temperatures can vary from the low single digits (celsius) to the upper 20s (celsius). To we Americans, that means from the mid-30s to the low 80s (fahrenheit).
So, the two-hat solution, combined with the variety of other gear I carry, enables me to adapt to any weather conditions I encounter. So far, that has included about everything except another Great Flood...of Biblical proportions...
I hope this helps.
Happy pilgrim me! 
