I've managed to narrow down one set of great-grandparents which is helpful. Sadly not with the 580 match yet - no tree for them and no common surnames between their matches so it's a bit of a haystack but process of elimination and logic is creating some clues.
One of the things you could try is starting a tree in Ancestry for these great grandparents and building downwards to present day as far as you can, then insert yourself into it at the appropriate level. You can do this by adding a fictional child for the great grandparents and making yourself descend from them. Then if you link this tree to your DNA results, Ancestry should go and find other matches with common ancestors related to these great grandparents. If you get a few more hits from this, then it probably means you're on the right track.
The matches themselves will see this as well but the benefit of making yourself descend from someone fictional is that living people can't be caught unawares at someone mysteriously appearing in their line - although if the 140cM matches are 2nd cousins, they're less likely to be concerned, but you never know.
Do the 140cM matches descend from the same or different offspring of the great grandparents?