Modern cycling helmets are well ventilated which is great in the summer but not when it’s colder. A basic woolen hat will more than likely be too cumbersome to wear underneath but thankfully there are several different cycling specific hats and caps for use under helmets. Skull caps may look odd but they are very warm despite being thin and the ear flaps keep you warm without hindering hearing. If all else fails then a balaclava will keep your face, neck and head warm on the worst of days. Traditional cycling caps are great for all year round use under helmets as they stop sweat, the visor keeps the sun or mud out depending on season and are fairly good at keeping heat in as well.
Off the bike you can show your brand loyalty as well as keeping your head warm with a range of beanie hats or baseball caps in lots of different colours and styles.