Sorry I did not see this until now but for future readers:

At the Hyatt Regency HLE the cars are stored in the hotel's valet garage. You can sign for your rental during office hours but you do not have to pick up the car until you actually want it. So you do not have to pay to park the car overnight. You can go and sign at 5pm or whatever time and they'll give you a slip for the car. You present the slip to the hotel valet at any later time and they will bring you the car. So you can come back early in the morning to get the car when you actually need it.

It is a great way to save on one night's parking in Chicago.

Charles