Does anyone know of company or driver that would get me from Civita to Spoleto?
Try Googling "Taxi Bagnoregio" and following the likely-looking links. Bagnoregio is the populated town near Civita. Be aware, though, that the distance to Spoleto is about 50 miles, so you'd be paying for a 100-mile taxi ride.
You can get from Orvieto to Spoleto by train in 1-1/2 to 2 hours, and Orvieto is linked to Bagnoregio by bus. Neither of those services is particularly frequent, but it could be done on a weekday (not on Sunday for sure).
There's also an all-bus routing via Viterbo, but that may well take longer than the bus/train itinerary via Orvieto. It would depend on how the schedules mesh.
I've been to Civita di Bagnoregio and do not consider it worth the cost of a taxi to Spoleto. Perhaps if you were squeezing 4 people into the taxi.