I am not sure if DL is willing to cover your ground transportation post-flight.
There is an easier solution. Take the refund from DL, and fly with a different carrier that would get you to EWR within your time desire - e.g. UA has an one-stop connection that would get you to EWR before 430pm (1630) - $119 Y (not basic economy) / $209 domestic F, at the time of this post.
With airlines consolidating their schedules, I'd look to making the experience as smooth as possible. Taking a car from JFK to EWR, only so you can drive back to SWF is not an ideal solution even if DL pays for the car ride between JFK and EWR. Right now you are relying on the airline to possibly make certain accommodations; instead, you can change the equation, so you are in control of your own accommodations along the way.