can anyone recommend a good taxi company to get from Florence to Cortona?

Cortona's over 60 miles from Florence. That seems like some taxi ride! Private driver, maybe?

My Google map route shows it's 133 km from the airport in Florence to Cortona, so I imagine a cab ride would be quite expensive. Plus, the cabbie is going to want to be paid for the return trip, too. Timewise, a train is going to get there just as fast.
My wife and I are planning a June visit, and are renting a convertible in Florence for two days to drive to Cortona to visit Siena and some other hill towns.

Flying in to which airport? I'm assuming Florence from your message. First, there are no train stations at either the Florence airport or Cortona. Aside from a private driver or taxi, which will be quite expensive, you are left with a taxi or ATAF-Voiainbus from the airport to the main Florence train station, a Regionale train from Florence to Camucia and then a bus to Cortona (about 25E-30E per person). You may not like the bus, train, bus journey to your hotel but it's a lot less expensive than a private driver. The only thing that could make this less expensive is to be in a group of 3-4 where the 120E bus/train fare begins to get in the range of a private driver.

Have you asked your hotel about what options might be available to you? They'll probably be the best resource. Good luck.

I have used AUTONOLEGGIO PRATESI S.A.S. for a car and driver to get from Florence out to the countryside. It's limosine service You can email them for a quotation - they have a website. http://www.noleggioconautistafirenze.com/ I emailed in English and got a reply in English. I used them twice about 5 years ago and recently inquired about another trip. They were reliable each time and the driver was very nice - spoke enough English to be understandable. I don't know how much it will cost. You will have to ask them.