Though I agree the timing of the request is kind of odd given the circumstances, I can say that I wouldn't be comfortable hanging up my own coat in First/Busine

Though I agree the timing of the request is kind of odd given the circumstances, I can say that I wouldn't be comfortable hanging up my own coat in First/Business, as the coat closet is under the purview of the FAs and not really a place for passengers to be going through. So the timing is weird, but the request in and of itself is totally normal for that class of service.