MSY has a consolidated rental car facility, but unlike many airports, no shuttle is required. It's a 5 minute walk (outdoor covered walkway) from there to the terminal, and the D concourse that Delta uses is the closest.
I don't have much experience with the D checkpoint, but I can't imagine with both PreCheck and elite status that you would have much of a wait that early on a non-Monday. 5 am arrival at the car return would likely have you at the gate by 5:20.
This seems like a rare case that I would recommend a rental car. If you do opt for a taxi (or Uber) instead, try to arrange for the ride the night before, especially if your hotel isn't right downtown or French Quarter (as free parking suggests may be the case).
Last edited by swag; Mar 16, 2017 at 7:09 am