Well, sort of. The continuation flight might use the original number, but the original flight must keep it unless it is cancelled.
Lets say flight 621 diverts on 25th and you send a plane to continue it, which will only get there on 26th. If flight 621 normally operates on 26th from A to B, it will have to keep the number 621, because that's what there are tickets for. The continuation flight from C can, however, also use number 621. It is departing different airport, and if it will still arrive at B well before the one scheduled for 26th, that's not a problem. See also Can an airline schedule two same-number flights to leave the same airport on the same day?. And if flight 621 isn't scheduled on 26th, well, then the number is free anyway.
That isn't rewriting the origin. It is a different flight with the same number in either case. And more likely it will get some disambiguation character, so it will be flight 621A or 2621 or such.
It is also unlikely for a domestic flight in the USA to be continued on the next day at all. There is enough air traffic that it's unlikely you wouldn't be able to get the passengers on other flights within a day if you can't get or charter a replacement aircraft for a day.
For an international flight there are such cases though. If you fly from South Africa and have to divert to Ascension Island, well, there is no other option than to send a replacement plane (that will also bring mechanics and spare parts to fix the first one) and it is likely to take a day to get it there. And it could even happen that the flight on 25th that diverted was full, but the flight on 26th is half empty and the smaller airline does not have another plane available, so it cancels the flight on 26th, rebooks the passengers from 26th on other flights and sends the plane to pick the stranded passengers from 25th, operating under the original flight number. And then they sort of rewrote the flight's origin, though it's really still rather a different flight with the same number.