General aviation Check Airman reported a near miss with another aircraft in the traffic pattern at a non-towered airport. The Check Airman maneuvered aggressively to avoid a collision then re-entered the pattern for a landing.

Narrative

Location: HWVOur aircraft: Aircraft XToday I had to execute an evasive maneuver to avoid a possible midair collision with another aircraft (an Aircraft Y). The maneuver was very steep and aggressive descending right turn with full power; we were at approximately 500 AGL at one point before we cleared of conflicts and climbed back to altitude.DETAILS: HWV is a Class G (non-towered) airport with an airport employee physically working at the main terminal who provides service such as Radio Check and advises aircraft of runway in use.HWV airport has Runways: 6/24 and 15/33.(All runways are left traffic pattern).We called ahead on CTAF asking 'Radio Check and Advisory'; At the time of the incident; Runway 33 with left hand pattern was in use (as advised by the airport attending advisor).At approximately XA:15; the candidate and I were at approximately 5 miles south-southwest of HWV flying east to west; maneuvering for left downwind entry for Runway 33 to conduct pattern work at HWV; we were at approximately 1;200 MSL.Our aircraft is an organization-owned aircraft equipped with Garmin G1000 with traffic; we noticed an aircraft on the our MFD (Multi-Function Display) was going opposite direction (the aircraft was going west to east slightly above our altitude; south of us along the Long Island shoreline (Long Island; New York)As our plan was to fly wide and away from HWV for proper 45 degrees entry for mid-field left downwind entry for Runway 33(the plan was to fly further west; then north; then continue right turn for the downwind entry).We made numerous position reports and announced our intention on CTAF.The candidate was the Pilot Flying; I monitored the environment with both G1000 MFD traffic indication and outside the aircraft; hence I had my eye on the parallel and opposite direction aircraft who was south of us.Then I noticed the aircraft made a turn toward us (they were going northeasterly direction at that time; while our aircraft still going east to west); closer and closer and dropped to our altitude; I had to immediately took over the flight control from the candidate; full power; nose dive toward the ground while right-turning to avoid the aircraft who was coming toward us from our left. It was a very aggressive evasive maneuver; the bank angle was close to 60 degree descending right turn; we were at only 500 ft. AGL above local residential area at one point; the other aircraft flew past us above.After clearing the conflict; we climbed back up to approximately 1300 MSL (local elevation is less than 100 ft MSL); and then at that point I heard the offending aircraft talking on CTAF announcing that they are going to Runway 6 and will be executing a missed approach. Since I am an instrument rated pilot who is familiar with approaches at HWV; I knew immediately they were doing HWV VOR Runway 6.I do not recall whether the aircraft made CTAF call.However I did query them on CTAF mentioning something like: 'Aircraft Y who is going to Runway 6; you were flying right toward me man.'The warrior replied the NY Approach Control turned them that heading and cleared them for the VOR Runway 6So; the story is that the pilots were not doing the practice approach on their own; they were with NY Approach 120.05 (the Class C Approach control; west of HWV)I submit this report; not to blame anyone; not Approach Control; not the aircraft shooting practice HWV VOR Runway 6But the following reasons:#1: I was very low altitude at one point (approximately 500-600 AGL above the residence south; southwest of HWV) with full engine power during the evasive/escaping maneuver; I was the pilot on the control during this sequence.#2: When Runway 33 at HWV is in use (again; all runways at HWV are left traffic patterns); HWV VOR Runway 6 would take aircraft flying toward aircraft that are in the traffic in the pattern of Runway 33 in the opposite direction (not directly head on opposite direction; but it's just a slightly off angle if I am in the pattern).Especially when the pilot intends to perform the published missed approach of HWV VOR Runway 6.(The approach; on the inbound would take aircraft flying toward Runway 33 traffic pattern. The published missed of VOR 6 will take the aircraft again interfering with Runway 33 traffic pattern of which the aircraft can flying toward traffic that are on left-base or final runway 33).Extremely dangerous; the pilot today who was practicing VOR Runway 6 did perform a published missed approach procedure; but I believe the aircraft discontinued and departed; perhaps they too realized the published missed would interfere with Runway 33 traffic pattern.