Single Engine low wing pilot reported a NMAC while in the traffic pattern at a non-tower airport.

Narrative

During a busy time at the airfield (S43); I entered downwind on a 45 for [Runway] 33R after having to do a 360 to make room for an airplane that had done a go-around and remained in the pattern. I made a call on frequency when I first entered downwind; and then again when I was midfield; calling out that I was #2 to land behind Aircraft Y (The plane that had done the go-around). Behind me Aircraft Z joined the pattern on downwind from a 45. I believe I heard him say he was #3. It was very congested on frequency; with several airplanes in the pattern; and others trying to get weather info; and other airports sharing frequency. I waited until Aircraft Y was abeam my position before beginning my landing sequence; and then waited until Aircraft Y was over the numbers of [Runway] 33R before turning base. I announced base; and then announced final when I began the final turn. The Aircraft Z pilot began his base turn before he had a visual on me; his call on the radio said 'looking for the Aircraft X' (I was announcing as Aircraft X). He completes his base turn ahead of mine and if he had continued onto final we would have been extremely close to each other. He must have seen me finally; because he took evasive action before turning final; and executed a 360 on the base leg; about 300 ft. vertically from my aircraft. I landed without further incident; and so did Aircraft Z. According to the ADSB data in my airplane Aircraft Z was about 300 ft. from me; but it looked a lot closer; and this could have been a mid-air collision. Pilots should confirm aircraft in the pattern ahead of them visually before making pattern turns.