I like the attitude Chancey displays, thinking he's in control of the situation.
The story the old woman tells about finding a ring previously is an oddly long monolog in the middle of this short story. If one doesn't know the history of the chiffoniers, they probably wouldn't get from a casual reading of "rats" that the rag-pickers were regulated by the city government. They were allowed to pick through the trash for usable items, as long as they kept out of the way of "respectable" folk and turned anything of real value over to the local authorities.
Such as a diamond ring.
So this would explain why she's so enthralled by the thing: she'd had one in her hands before, but was required by the laws of the city-folk to relinquish it, even though it would have been lost forever had she not found it.