PA-32 pilot reported an NMAC with an opposite direction aircraft during departure from a non-towered airport.

Narrative

Departed ZZZ on direct flight to ZZZ1; VFR. After takeoff heard position report of local RV-7 on assumed VFR approach to ZZZ - position report indicated '10 miles South of ZZZ airport; inbound for Runway XX'. My position report indicated 'Departing ZZZ to the south-east'. Position reports indicate separation of several miles and that aircraft should not be a factor. Approximately 5-6 miles from ZZZ at 2000 ft. the inbound aircraft passed directly in front of us on descent into ZZZ. No evasive action was required. Neither I nor a passenger with me who also holds a PPL saw the aircraft until it was almost directly in front of us; passing left to right and descending. Discussion with the other pilot was had and we both agreed that the other plane was obscured by the ceiling and A-pillar of the aircraft windshield until it was almost in front of us. I leveled at 2500 ft. and continued on my flight. I called ATC about 2 minutes later to obtain clearance into the ZZZ1 Class C. Approach made no mention of the near-miss and I do not believe that they noticed it. I did not mention the near-miss to ATC but upon landing and researching NMAC regulations under section 3 determined that I likely should have asked to make an NMAC report. After discussion of the situation with my CFI it was decided to make this report and to also alert the FSDO when they open in the AM to file a proper NMAC report.