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Upvote:4
Typically the bag drop after customs is designed for bags that are already checked through and tagged, which normally requires this to be a single ticket. It's easy enough to check: if the tag on the bags say JFK you need to go to Delta check in counter and have a new tag put on. If it says ATL, you can simply drop it off directly after customs.
You will also need a Delta boarding pass, but you can potentially check in online with an app.