The flight instructor reported that during a public-use flight in a single-engine airplane, he was providing instruction to an airline transport pilot who was only rated for multi-engine airplanes. The flight instructor further reported that the pilot receiving instruction was the pilot flying and was "fast" on final approach. During the landing flare, the airplane ballooned and bounced once. After the bounce, the flight instructor applied aft flight control pressure because he believed the airplane was going to touch down nose wheel first. The subsequent landing roll was completed without further incident. After loading additional passengers, the flight instructor completed the return flight to their domicile without further incident. During a post-flight inspection damage was found to the tail hook and aft fuselage area. Further inspection revealed that the airplane sustained substantial damage to the aft bulkhead, which was likely the result of a tail strike during the bounced landing during the previous flight. The flight instructor reported that he did not hear the tail strike during the bounced landing and he did not inspect the empennage before departing for the terminating destination. The flight instructor did not report any mechanical malfunctions or failures with the airplane that would have precluded normal operation.