The thing about an under-the-helmet head covering, whichever of the many good suggestions you choose to run with, is that it is a much greater effort to change it mid-ride if you find out you didn't dial in perfectly from the start.

For that reason, I'm a huge fan of the helmet cover, which is super easy to whip off mid ride (albeit a bit more of an effort to put back on).

Once it goes below about 50, I almost always leave the house wearing one. If / when I take it off varies. If it is very cold, I might wear a skull cap under the helmet and still throw on the helmet cover over. More flexible to have the two layers, one of which is easily tweaked.