Woburn Park and Ride Overflow Lot?

Hi all, I live in Maine, and am flying out of Boston next week on Tuesday. My plan is to drive to the Woburn Park and Ride, leave my car there, and take the shuttle to the airport. I'm a little worried about the lot being full tho, on account of the holiday. Is there an overflow lot, or other nearby parking I could go to? Additionally, does the Massport site post if the parking lot is full, so I can find a backup plan if needed?

Thanks