Hi,
I've found that if you upload a new extension and just save it as a draft that the 60-minute review happens in the background - when you actually come to publish the extension, the new version is available immediately. This doesn't work if the extension needs to go through a manual review.
I'd also advise that even if you set the extension to Public, it will not appear immediately in the Chrome webstore search results, so it might just be worth publishing as Public slightly earlier anyway.
You don't need to unpublish and then republish to change visibility.
Cheers,
Simon