Small aircraft pilot reported conflicting traffic entering traffic pattern resulted in a NMAC. Pilot also reported this airport is non-towered yet very busy with frequent traffic conflicts.

Narrative

I announced my takeoff on Runway 02 and intentions for a downwind departure to the south. I recall hearing Aircraft Y on 122.8 (CTAF) performing a left 360 over Hayden Lake waiting to enter the pattern. After departing from Runway 02 at COE; I climbed straight out to 3;000' for spacing before turning crosswind due to multiple planes in the pattern. Upon announcing and starting my right crosswind turn I saw another aircraft out in the distance and appeared to be coming towards the downwind leg. I continued as normal while monitoring that other aircraft which did not appear on ADSB. On my crosswind leg I noticed that Aircraft Y coming towards the pattern which I felt required more awareness due to conflicting flight path and altitude. At that time; I announced my downwind and I continued climbing to 3;450' for added vertical separation and Aircraft Y passed just in front of and maybe 300 feet below my aircraft. I lost sight of Aircraft Y as the nose of my aircraft blocked the view until the aircraft was further to the west before regaining visual. Aircraft Y passed under the traffic in the pattern below pattern altitude. I am unsure if Aircraft Y was aware of my position or not. I have reported this to the COE airport. The Coeur D' Alene airport has been becoming increasingly busy and is a dangerous place to fly with no Control Tower. I had a few other close calls with traffic during my primary training at that airport as well. COE has increasing private jet traffic; general aviation; and flight training. Many people have had close calls their as well. It's ongoing issue and an accident is waiting to happen. A temporary Tower has been built but this airport truly needs some attention to improve the safety. I would love to see the temporary Tower operational year round during busy daylight hours.