Skymark Airlines is one of the few small independent airrlines and has flights between Tokyo and Kagoshima, Fukuoka or Tokushima.
Competition within the transport industry
Transports are dominated by 3 companies, which basically set the prices of transport in Japan. These are the two national airlines (JAL and ANA), and the Japan Railway. Domestic flights fares are set just a bit under the Shinkansen (bullet train) prices, while the latter are in turn set just under the cost of the journey by car (including fuel and highway fees).
This explains for the unreasonably high cost of domestic transport inside Japan compared to other developed countries. A flight from Tokyo to Hokkaido or Okinawa can now cost as much or more than a Tokyo-London flight, and is always pricier than a Tokyo-Seoul or Tokyo-Los Angeles flight.