Sometimes its going to be hot and sunny, and sometimes its going to be cold.
You’ll need two hats one for the heat and one for the cold
For the cold:
A beanie style hat made of wool or other warm material (such as polar fleece) that will retain heat even when wet.
You may have a jacket with a hood built in but a beanie hat can got under this and you will be even warmer.
A scarf in addition to the hat is also a good idea.
For the heat:
usually when it’s hot the sun is out and a hat can also protect your skin from sunburn so a brim on the hat is ideal.
A baseball style cap will provide some protection but a hat with an all around brim is better, a bucket hat with a small all round brim is good.
Cotton is a good material as it allows sweat to be absorbed and then wicked away and evaporate. when stored cotton survives being crumpled.
Remember to protect your neck when it is sunny.
$10-15 from many places