I have the same problem. My flight is 10PM. I signed up for Port Valet. The confirmation email I got said:
*If you are flying Southwest Airlines, please note that your departing flight must be between 11:00 AM to 4:00 PM on the day of the ships arrival.
Unfortunately, my flight is with Southwest.
I found another solution. I am going to take my luggage to the airport and store them there. There is a service by SmarteCarte. It is located on the arrivals level of Baggage Claim 9. There is a light rail that connect the airport to the city. Seems to be quite convenient to do all the sightseeing without a car.