Not necessary to change hotels, though you can, if it makes you feel comfortable. You should know that on your return to the airport, Super Shuttle guarantees on time arrival. I lived in LA for 50 years and I have left for the airport at almost every imaginable hour, and I have never been even close to being late. The airport from your location is really a direct route down La Cienega. No freeways required. It really is quite fast, especially during early morning hours.
However, if you don't mind spending the money for parking, which will be about 30 dollars a day, a rental car is far and above your best choice. That would give you a lot more freedom to come and go. If you want to rent just a few times, Enterprise is just down the street from the hotel. Hope it helps, ANY other questions, don't hesitate to ask.