If you actually talk to the teachers themselves and not the receptionist, some of them have unpredictable enough lives that they might be willing to meet you on a few hours notice when schedules match. You could almost make a habit of calling at 4pm on days you are free to see if there have been any cancellations or holes in someones schedule.
Private lessons are going to be expensive. You are hiring an employee for an hour, someone who has to make a living from an incomplete schedule, so the cost for an hour can be nearly $100.