One thing to be careful of is that livery service generally requires a particular licensing from the local gov’t (at least in the US).
So, while you CAN hire someone to drive you around in your car, they might be doing so illegally.
That said, I’d look at what a reasonable price for the time would be, and convert that to a weekly or daily fee, paid twice monthly. E.g., if you can convince someone to do it for $10 an hour, and it takes them 30 minutes to get to your house, an hour to drive you, and 30 minutes for them to get home twice a day, your daily rate is then $40, weekly is $200, and monthly around $800 (assuming a 20-day a month work schedule). Paying weekly/monthly means that you don’t have to pay them extra for heavy traffic, and they make extra if they can make it more efficient (or on light traffic days).
I think $10 is a little cheap, but it depends on where you are. Here in Pgh, I bet I could find someone (probably a college student) to work for $40 a day and keep their schedule cleared at the right times.