Sorry for the delay.
I hired a car service. It was much less expensive than working through the hotel.
I found the company on trip advisor. The main contact is Hector (email below)
[email protected] OR
[email protected]
We used them three times and generally were satisfied. The cars were clean and the drivers very kind and hospitable. Cash payment is preferred (pesos or USD). For one ride I arranged in advance to pay credit and their machine was not working.
I was less impressed with their ability to provide tour guide type services (as advertised on Trip Advisor).
For the drive from MEX to Puebla I was happy to be in a suburban or SUV. That seems like a route some drivers are tearing through. That said I was with two young kids and frankly there wasn't any issue with reckless drivers.
From and within Mexico City, uber is fantastic. You have car class options (compact to suburban). They come immediately and are very reasonably priced. (Hector is probably better for airport pickups where someone can come in and help you with luggage etc.)
In Puebla, uber was great as well but there are no car size options so you'll almost always get a tiny compact picking you up.