I know the coffin corner is only meant for subsonic aircraft, but if the Concorde is just cruising at Mach 1 and had an engine failure and also lost that afterburner, would it enter a coffin corner situation?
The Concorde doesn't need reheat to cruise supersonically, just to get there (the range would be pretty short if it did). If it loses an engine it's not going to put the other 3 into reheat to hold altitude and speed (fuel burn) and it will do what just about all airliners have to do when at they lose an engine, descend to whatever its engine-out service ceiling is. This is done at some speed near minimum drag that gives the flattest descent, called "driftdown" speed, whatever it is, and it will just pitch down as required to maintain that speed with the available military rating (non-afterburner) thrust. Obviously, it will come out of mach as it does so.
Also I should point out that being a supersonic aircraft, it doesn't really have a "coffin corner" in the normal sense. That applies to subsonic airplanes that are limited to a speed just shy of Mach, and when they get so high they become hemmed in by stall speed on one side and trans-sonic related issues on the other. The upper limit of a supersonic airplane's speed is aerodynamic heating related and is a function of the airframe material's ability to run at high temperature, which limits aluminum structures to below Mach 3. This is why the SR71 is titanium.
The coffin corner is the altitude where your maximum speed (limited by high speed buffetting) is equal to your minimum speed (limited by low speed buffetting / onset to stall).
The Concorde, when cruising at Mach 1, is not in the coffin corner, regardless of its engines health. Both the upper speed boundary and the lower speed boundary* of the flight envelope are not affected by the engine health.
*when taking non-linearity due to high angles of attack into account, the engine thrust has an effect on the minimum speed bound of the envelope. That is not relevant to the coffin corner situation, because of limited angle of attack.
Coffin corner is about speed: the small margin between stall and critical mach.
You don't need engines to sustain speed. You can use descent instead of engines.
As you descend, the margin widens. Soon you find an altitude where your remaining engines can power you at a viable speed.