FWIW - I've read years ago that the Caller ID info is sent after the first ring, but I believe it may sometimes be sent a little later. I don't believe any phone network/system sends the data more than once, so if the data gets corrupted - it's history. Also, the Caller ID system between various phone companies within the U.S. is not always compatible, especially 100% compatible.
My 622, my table phone with external Caller ID Box, and my Cordless Phone with Caller ID, all seem to work together; but they don't always obtain the Caller ID info after the 1st ring, sometimes it's the 3rd ring. I have service with QWEST.
At various friends homes:
...(1) Caller ID would often not work on some phones when there were too many phones and answering machine on the line. I suspect a voltage drain causing the Caller ID info being corrupt.
...(2) At a second friends house, when Katrina passed by, there was flooding and it was shorting the line that went below a floor. The phone worked with static, but it corrupted the Caller ID.