General aviation pilot reported a near miss with another aircraft in the traffic pattern at a non-towered airport. Pilots of both aircraft maneuvered to provide separation; the general aviation pilot departed the traffic pattern for a different airport.

Narrative

I departed from FFC runway 13 in a rented aircraft for a personal refresher flight to fly a right traffic pattern and return to the runway for a touch and go (right-hand pattern indicated for runway 13 at FFC). There were several aircraft in the pattern; before I took off; I heard radio calls and saw on the ADS-B one aircraft turning downwind to base; and another aircraft in the downwind near midfield. After announcing my departure and plan for a right-hand closed traffic pattern on the CTAF; I took off; past the departure end of the runway; I turned right crosswind and continued my climb to pattern altitude. When I was at roughly 1;700 feet MSL (roughly 900 feet AGL) I was preparing to make a right turn to enter the downwind leg and level off at pattern altitude.As I was starting the turn; I noticed Aircraft Y approaching from my left at pattern altitude. I quickly descended to maintain vertical separation while the other aircraft conducted a left 360 to re-establish horizontal separation. I announced on CTAF that I was remaining below pattern altitude until horizontal separation was re-established; and the pilot of the other aircraft angrily announced the left 360. Once separation was re-established and I was established on the downwind I decided to change my plans and conducted a downwind departure to go to a less busy airport for my touch-and-go practice. While departing; Aircraft Z came on CTAF and accused me of cutting off both him and Aircraft Y in the pattern.I do not recall hearing any CTAF transmissions from either aircraft and did not observe either one as being in the pattern on ADS-B prior to my takeoff; but it is possible that they made calls that I missed while focusing on the other aircraft in the pattern. Once I began the takeoff roll I was focused on the engine and flight instruments and did not look at ADS-B traffic again.I do not know the flight path of either other aircraft; but it seemed to me like they were entering the pattern via a nonstandard entry - entering the downwind leg from the departure end of the runway; instead of at a 45-degree angle at midfield. I can't think of any other flight path that would have led them to be flying opposite runway heading; at pattern altitude; in the vicinity of where aircraft would be turning from crosswind to downwind.The FFC airport is growing increasingly busy and I hope that a control tower and class D airspace can be established as soon as possible before a collision occurs in the traffic pattern.