I thought i'd seen it somewhere...
Could just be a coincidence and there are probably quite a few hats in there, but...
Descent (150) oct/nov 1999
Gives an article about the Explorers of Eldon Hole. (pg. 26-27)
on page 27 in the middle column is a short exert, which begins 3 lines up from the bottom.
"when a man in ascending, dropped his cap into the hole, our friend shouted a rigmarole, of which the only words we could catch were: 'winterbottom - coming up the hole - cut - his head', 'cut his head?' we enquired sympathetically, and a wrathful shout was the reply.....somebody upstairs came to the rescue, and explained that the word was 'cap' and not 'cut', and our fear of a sanguinary accident was relieved"
Could this be Mr Winterbottom's hat?
Also, this article gives some pictures of the use of the bosun's chair, and one of the several walls which have been seen around eldon in years past.