To get the Regalia Type-F you have to complete the game first. After the end credits, save the game and load it up again. You will be back in chapter 14 where you have to go to the beds, call Umbra the dog and return to Lucis.
After returning to the open world, you have to complete the Imperial Base called "Formouth Garrison" in the north.
It is very tough and you should be at least lvl 40. After completing all waves you will get the "Strange Engine" Part for your new Regalia Type-F.
Go to Hammerhead and claim your new flying car at the garage.