So here is how many missiles increase range. They take an upward trajectory initially, and then use gliding to increase range. They then use gravity to increase speed as well. As you increase altitude, air becomes rarified, resulting in less oxygen available for combustion. Which is why missiles carry an oxidizer.
For an air breathing missile, you need to show that combustion is so efficient it can give you speeds of Mach 4 over long distances. I am not aware of any fuel that would do that, but then again, my knowledge in this area is limited.