Did you book online? If so, you can use ATA's express check-in. These are kiosks where you scan the credit card used for the reservations, answer the questions regarding checked luggage, and obtain your boarding passes. An attendant then puts the luggage tags on any checked bags and places them on the conveyor belt. You're then off to the gate.
This is really convenient, since the wait to check-in can take 30 minutes or so for those first flights in the morning. ATA has had express check-in at Midway since the beginning of the year and we have used it three times. One time we had to wait about 10 minutes, but the line for the regular check-in was at least an hour long at that time (spring break traffic). On the other two trips, we walked up to the kiosk, completed the check-in procedure, and were on our way to the gate in about 5 minutes.
I would still allow at least an hour for a 6 am flight from the time I enter the terminal to flight time (allow more time if you are parking in one of the $9/day lots, since the shuttle buses take a little time to get around and don't seem to be as frequent that early in the morning). This is likely to be the first ATA flight of the day outbound so it won't be as busy as it gets later in the day.
Ralph