If you read the label on chicken feed, it will say something to the effect - "this is a complete feed, no supplements are necessary." That means, feed chicken feed and nothing else.
When foraging, they will supplement a small portion of their feed but that won't happen between October and March when everything is dormant.
Normally you don't have to bump up the protein. If they're molting, they're not laying eggs and need to build feathers. During that time switch to a higher protein starter/grower feed.
During the laying season they only need layer feed. Providing oyster shell in a separate container is a good idea.
Giving corn free choice will allow the chickens to eat all they want and they won't get the nutrition they need.