Basically, yes, you can, but only under certain circumstances.
First, you'd need to schedule an appointment at the passport agency using their toll-free number (as I remember from the time I was there, they do not like walk-ins, and if you come without an appointment, there is a good chance they'll turn you away).
See
http://travel.state.gov/passport/npic/agencies/agencies_913.html
It is OK to call before the oath date but, obviously, the appointment itself needs to be after the oath.
At the passport agency they'll ask you to show proof of travel within 2 weeks from the appointment date.
How quickly the passport will be ready depends on how much time you have between the appointment date and the travel date. Generally, the Chicago passport agency issues passports in 0-3 business days. They do issue same day passports, but only if your travel date is really close to the appointment date. In my case my appointment was on Friday, and my flight abroad was on Sunday, so they did issue me a same day passport. But a friend of mine had about 10 days between the appointment and travel dates; his appointment was on Thursday and his passport was ready on Tuesday of the following week.