I handle the lens cap in different ways... If I am carrying a bag I have a space reserved for the cap. If I am wearing a shirt with a breast pocket or a photo vest, I have a pocket reserved for the cap. I try my best to keep the cap free of lint and pocket grime so, when I am shooting in a tee shirt and a pair of jeans, I have this on my belt...
https://www.ebay.com/itm/KADEN-Water....c100005.m1851
It keeps my cap clean and I always know where it is. Cost less than 2-dollars too - which is cheaper than replacing a lost cap.
Funny thing about lost caps! At one event, a guy came up to me handed me a Canon cap and said "you dropped this". I thanked him and popped the cap into the pocket of my vest. When I reached into the pocket to put the cap back on the camera, there were two caps in the pocket! I realized that the guy had handed me someone else's Canon lens cap which was the same type and size as the one I was using...