You'll definitely get plenty of different opinions on this subject. I'll offer my thoughts on it. Just keep in mind that a good deal depends on your style of riding. Myself, I ride slow enough that I'm not all that interested in the helmet for safety's sake. For me, it's more a matter of comfort. During summer I don't wear a helmet at all. I'm more interested in protection from the Sun and keeping cool. So I wear hats that address those issues.
But during the winter, I wear this:
A very inexpensive bicycle helmet. I filled the breather holes with silicone and attached a bill. Attached a short visor to that. It's waterproof.
Underneath it I wear cheap stocking caps. The ones that you'll find in the boy's clothing dept at a big box store. Kinda small and thin for a grown man who's using it as just a hat. But they're just right for getting underneath a helmet.
The combo is plenty warm. I've ridden in single digits, F, without my scalp feeling the slightest bit of coldness.
Do keep in mind, though, that I average about 12 mph. Requirements will be different for anyone who rides faster.