GA pilot reported an NMAC while turning to enter the traffic pattern when an aircraft flew directly below them without announcing their position. The pilot made an immediate climb to avoid the aircraft then returned for landing.

Narrative

I experienced a near miss on entry to the traffic pattern with another aircraft already in the left downwind. Coming from the east; I made a planned an entry to left hand traffic for [Runway] 16 via flying over midfield at 4;500 MSL (1;500 feet above traffic pattern). During descent from the east to the ZZZ airport I made standard calls at 10 miles; 8 miles; 5 miles; and 2 miles. No response or traffic callouts were heard coming from ZZZ; but there were calls from the ZZZ1 airport nearby on the same frequency. Additionally; TIS /ADSB on the G430 and iPad with stratus; showed traffic at ZZZ1 ; but none at ZZZ. Upon crossing over the field and and extending 1.5 miles to the west; I considered it safe to turn downwind as I perceived that there was no one in the traffic pattern (no traffic calls; no TIS/ADSB; no airplanes seen visually). When I was about to enter downwind traffic and make a left turn; I was very surprised to see an aircraft within 500 feet directly below me. Our paths crossed but fortunately enough separation was maintained to prevent a mid-air collision. I immediately added power for altitude and flew away from the field. It was then that I first heard a radio transmission from the other pilot giving a position report. After establishing communication with the other aircraft a normal landing was completed. Fortunately no one was killed in this situation but it very well could have been deadly if my traffic pattern entry was non-standard or if the other airplane was flying a high pattern. My aircraft had lack of visibility downwards and the other aircraft had a lack of viability upwards which contributed to the situation as well.Communication is what I feel could have prevented the near miss in the first place. Whether it was a radio issue or a silent pilot in the other aircraft; radio calls were not received on my end; and along with no ADS-B; contributed to my perception that the pattern was safe to enter. In the future; I will fly my entries to non-towered airports 1;000 feet. above pattern and extend 2 miles away before turning to join downwind as described in the AIM in order to provide a larger buffer of safety. Additionally; I will increase my scan for traffic when entering to see and avoid other aircraft and stay vigilant even when ADS-B and radio calls do not indicate traffic.