If it's very cold, however, try to wear some sort of head gear. The winter cold dries hair out. Your strands of hair don't like floating through cold air freely. If you jog outside, like I do, tie a scarf over your head before you hit the pavement.
I used to wear a scarf even in the summer because running creates resistance between your hair and the air but I've decided this isn't all that damaging. Having a TWA right now makes it even less concerning.
If you prefer to wear a scarf even in the summer, go ahead. You won't feel much hotter with the scarf on but it does provide some protection esepcially if you have very thin hair strands that break easily.
If you don't have a scarf available twist your hair or tie it back if it's long enough. Your strands fair better when they are bunched up together in a twist or a plait rather than when they "float through the air".
If you go to a gym, obviously this isn't going to be a problem. I usually prefer outdoor exercise and commit to it even when it's very cold outside. I have jogged in light snow, rain and at -4 degrees celsius, you get used to it.
When to wash your hair
I don't think you need to wash your hair after every single work out but I frequently co-wash with the Queen of Kinks Conditioner.
I don't get smelly hair (according to my husband) if I leave it unwashed after exercise for a couple of days but I know you might or at least just feel uncomfortable and self-conscious when you don't wash your hair after a workout.
If you prefer a wash after every single work out, co-wash rather than shampoo for a gentler clean and shampoo once or twice a week only.
If your hair is flat ironed it will get frizzy after a jog but that's not a problem for the many naturals that avoid heat.