Are your daily rates within a 24 hour period, and do you have overage charges for being late?

How do you get paid for the extra hours? I have a pet coming on a Friday 8:30 am to Sunday 6:30 pm (estimated), but the client only paid 2 days. The client has asked me to be flexible. I did tell the client that my daily rate is for a 24 hour period, so 8 am Friday to 8 am Sunday would be 2 day rate. I would be flexible within a couple hours after the 48 hour period but 10-12 hours later, I would like to charge a 1/2 day rate. She even said she may not come back until Monday, which is more than 48 hours she paid for. Wondering how I would charge her for the extended time ? I have a client that wants me to be flexible based on when they decide to come back. However, they only paid for 2 days. Suggestions in how to handle?