The term Hypersonic is generally accepted to be Mach 5 or above. Not simply above supersonic.

The rate at which objects travel faster than the speed of sound is expressed in terms of supersonic and hypersonic speeds. Such speeds are measured in terms of Mach. Mach number is defined as the ratio of speed of the aircraft to the speed of sound. Supersonic and Hypersonic are the two main types of speeds of flight. In scientific language, they are better known as the regimes of flight.

Supersonic speed

Supersonic speed is the rate of travel which exceeds the speed of sound, also referred to as Mach 1 speed. Since the speed of sound depends on temperature and composition of air, supersonic speed also varies with changing altitudes. In water at room temperature, it is greater than 1440 m/s, while in solids it is even more. An example of supersonic is, a bullet fired from a gun. Fighter planes and space shuttle also fly at these speeds. Concorde is the only passenger aircraft to travel at such speeds. It took its last flight in 2003 and it is not in use now. The term sonic boom is associated with these speeds. When an aircraft flies at this speed, a person on the ground hears a very loud noise, which is called sonic boom. Sonic boom is produced as a result of the movement of air molecules under a tremendous force.

Hypersonic speed

Hypersonic speeds correspond to very high supersonic speeds. They are basically Mach 5 speeds or five times the speed of sound. The speed of a hypersonic aircraft is around 3000 miles per hour. These speeds are again of three kinds, low hypersonic, hypersonic and high hypersonic. X-15 was the only manned aircraft to fly at low hypersonic speeds, i.e. at Mach 6. During its re-entry, space shuttle also acquires such speeds. The distance between shock wave, generated by a hypersonic aircraft and the aircraft is smaller than in case of supersonic aircraft.

Some good answers in this thread, so I'll just add the trivia that a jet can fire its gun, and then maneuver in such a way that it shoots itself down with those same bullets. Ballistics is a funny thing. Here's the story:

In 1956, a Grumman test pilot was flying the F-11 Tiger fighter jet off the coast of Long Island. He dropped his nose 20 degrees, fired a brief four-second burst from his four Colt Mk.12 20-millimeter cannons, and entered a steeper descent before hitting the afterburners. A minute later, his windshield caved in and his engine failed. The pilot had assumed he had been the victim of a bird strike, but the accident investigation revealed that he had actually flown into his own stream of 20-millimeter cannon rounds. Although the rounds had a head start, they slowed quickly due to drag passing through the surrounding air. The rounds decelerated, the Tiger accelerated, and the two reunited in the sky with fatal consequences for the aircraft. The Tiger was totaled during the crash, and the pilot, while severely injured, was able to return to flight status less than six months later.

I'm sure a jet flying at mach-1 is well above the safe threshold for releasing missiles. Just because they can fly that fast doesn't mean they can launch anything at that speed.

Fun fact

Trident II missile=18,030 mph

Nuclear subs carry them.

Mach 24

https://spaceflight101.com/trident-ii-missile-test-wows-observers-in-california/