If you want private, I suggest checking with the hotel concierge in Siena and see if they can arrange for you. We have used Groundlink for several European transfers. They are based in NYC, but worldwide. Had a cruise line recommend them years ago and they have helped us in 4 European locations. Excellent!
This may help …firenze.it/en/…limousine-service.html
I have not used this service, but I'm e-mailing them in regards to transportation. Their office is in Florence. Italy Chauffeu Service. They have been very prompt in their reply.
Try Tommaso's transfer service: http://www.greve-in-chianti.com/taxi.htm
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