I think you may be slightly misinterpreting the picture, Bill. In all the wooden barrels I’ve drai~, er, examined, the head boards are not restrained by any interior wooden hoop. They are set in a recessed channel run around the inside edge of the staves, a few inches down from the rim. This slot is always completely straight, hence our expression “as the croze flies”

Croze
Pronunciation: krôz
Noun: A groove inside the end of a barrel or cask into which the head is set.
Etymology: French creux, from Old French crues, groove, from Vulgar Latin *crosus, perhaps of Celtic origin.

The American Heritage® Dictionary of the English Language, Fourth Edition. Copyright © 2000 by Houghton Mifflin Company. Published by the Houghton Mifflin Company. All rights reserved.

I think the chime refers to the part of the staves which overhang the head boards.