The purpose of “~ㄴ/은 채” is to indicate that the state of the clause describing it continues until (and usually beyond) the action I the next clause. I want to stress the word “state” in that sentence.
When I say “모자를 쓴 채”, it does not mean that the person actively put on his hat. It’s possible that he put on his hat in an earlier sentence – but that information is irrelevant to our current sentence. All that is relevant is that the hat is currently on his head… and that state (the hat being on his head) will continue until (and beyond) the next action.
So… let’s finish that sentence. If I were to say, for example:
선생님이 모자를 쓴 채 학교에 들어갔어요
… that sentence would translate to: The teacher went into the school with a/the hat on his head
As usual, it’s hard to come up with an English translation that fits all scenarios. The most common translations are “with” or “while.”