So we wear a hat in winter to keep warm and a hat in summer to stay out of the Sun. This is rarely the same hat, unless you live near the Arctic Circle. So what do we look for?
There are a few winter myths that need clarification. We used to be told that an exuberant amount of heat was lost through our head. We actually lose heat from all parts of our body; the head might be slightly more sensitive, but not a lot. It just that the head is the part that is least covered up, and we lose heat from the area that is least covered up. Really, you have the same problem with not wearing gloved in cold weather as not wearing a hat; no glove means cold hands, no hat means a cold head. As hands and heads are both important we strongly suggest appropriate gloves and headwear.
Another myth is that we catch cold from being cold, or maybe being wet. If you feel really cold you probably are staining your health a bit, but the flu is caused by one of 200 different viruses, not the weather. Nonetheless, you can get hyperthermia from exposure to the cold; if you are cold enough to shiver you are taking a risk. Also, cold weather appears to supress the immune system (1), which means you are more likely to catch the flu if you do find yourself exposed to it. Being a little warmer with a hat and gloves is a healthy way to go.
So what hat to wear in cold weather? This s a stylistic matter, many different hats can keep you warm. The best materials are obviously ones that provide good insulation: wool or polyester. Design wise, find something that covers the ears; ears are exposed and don’t usually have a layer of hair over them. Other than that, find a beanie, a trapper hat, or a knitted beret. If you prefer more traditional hats, augment them with a pair of ear muffs. As some hats (like beanies) are both cheap and fairly compact it can pay to have a few on hand; keep a beanie in the pocket of every warm coat; sooner or later you will find yourself using it.