If a taxi/rideshare requires a shuttle bus, would it be easier to take the Airtrain to another terminal (e.g., it's very easy to get from the airtrain to pickup in T8) or maybe take the Airtrain to Lefferts Blvd and to save a few dollars. There's the recent thread on rideshare savings from Lefferts: JFK or Lefferts Blvd Airtrain? Ride share info - Fun Fact
There's discussion about issues with the current T4 bus on that thread too.