HCA Airport has partnered with Taxa Syd. Going from Odense to the HCA Airport, you get a 10% discount on the trip. Further, there is set a fixed price from the following cities:
Read more about how to book an airport taxi here. (danish site)
You also have the option to order a taxi directly at the airport when you get home. There are two kiosks where you can book your taxi and follow the on-screen update until it stops infront of the airport.