GA Flight Instructor with student reported an NMAC while in the pattern of CPT non-towered airport. The pilot of the other aircraft took evasive action to avoid a possible collision.

Narrative

I was conducting a flight with a student entering the CPT airport. We were entering from the east of the airfield to make a teardrop entry 500 FT above the airport TPA (Traffic Pattern Altitude) to enter the right downwind for Runway 15. Student make initial call of approximately 10 miles; 6 miles; 2 miles; crossing midfield; and executing teardrop to enter the pattern. My student and myself noticed there was another aircraft in the pattern (tail number unknown due to this aircraft not having ADS-B) so we were making sure to keep track of where this plane was. Again my student made multiple calls advising that we would cross midfield at approximately 2400 MSL; then left teardrop for the right downwind; Runway 15. As my student called that he was executing the left teardrop; the plane (without ADS-B) called they were turning final. As we were coming through the teardrop and entering the pattern; I was actively looking for any traffic and was unable to see anyone. Once we had entered the right downwind; I heard this aircraft call he was entering the right downwind. Almost immediately; I noticed the plane was turning from crosswind to downwind close behind our plane. Fortunately; the other pilot then saw us and stated he was going to make a left 360 for spacing for the aircraft in front of him (our plane). All additional radio calls were made; including downwind; base; and final. No additional communication occurred between myself or the other pilot. The rest of the flight was completed without incident. I believe this could be prevented in the future by making additional calls while in the teardrop maneuver and making an additional 360 degree turn if I am not completely sure where the non ADS-B traffic is at. As well as the other aircraft making additional radio calls for departure and crosswind legs.