It’s not the stiff, round sort of beanie that is rather like a yarmulke.
It’s a soft, knit hat, sometimes with a pompom, sometimes with a cuffed edge. Made for cold weather wear. Sock cap, tuque, knit cap, ski cap, toboggan, beanie, there are probably other terms.
I think sugar-free makes a difference - the sugar in the regular kind feeds the mouth bacteria, so you end up worse off than when you started.
Of course, you could always stop smoking and drinking coffee. 
I used to grow up thinking I had no BO whatsoever.
Then I had to move some furniture with a colleague a good 10 years ago.
I was breaking up a sweat and we then had to drive together about 150 kms.
He subtly motivated me to start using deodorant by yelling at me “Jesus, you stink, man!!! Haven’t you ever heard of deodorant ?!?!?!”.
I have never been so mortified in my whole life and since then I religiously apply deodorant every morning.
Where and how you smoke is a factor too. When I was a smoker, I tried to smoke outdoors as much as possible and I held the cigarette away from my body. I’d wash my hands after every cigarette. People at work were shocked to learn that I smoked.
I did notice that the smoky smell tended to cling to me more in cold weather.
“Yes, hello? Child protective services? There is a man with a child in the store where I am, and he is openly stinking right next to the child! No, I don’t know if he is the child’s parent or not, but he smells really really bad and he’s near the baby, don’t you understand?!?”