Originally Posted by MikeNYC1

Oh, and one thing you can do to save time (if the initial gate agent doesn't request it in TPA) is to bring your passports to the TPA gate agent on departure and ask that person to verify them (unless you're checking bags in TPA, in which case they'll do it then). That way, they can print you a boarding pass with "DOCS OK" on it for your JFK-ZRH flight, so you won't have to deal with that at JFK.

They'll likely request your passport in TPA first, but if they don't ask for it, you can proactively provide it. It'll save a few mins and ensure you can walk right onto the plane at JFK when you're connecting.