While I don't know the nuances of this with particular respect to flying Delta, I can give you what my experience has been with other airlines. Generally, it's at the gate agent's discretion if they will book you through on a connecting flight. If both flights are wide open, there's a good chance they will do this. I suspect you won't be as successful if you're connecting through a city that wasn't on the original itinerary and the connecting flight looks like it could fill. As for standing by for a later flight that ends up being the following day, it's highly likely you would be charged a change fee. Generally the fee is only waived for standby, same day travel.