Pilot on VFR cross country flight inadvertantly entered IMC and exited back to VFR conditions with ATC assistance.

Narrative

Flying VFR from ZZZ to ZZZ1 First leg ZZZ to ZZZ2 due to weather briefing; and monitoring on my electronic flight bag. Second leg ZZZ2 to ZZZ3 ; but assessment of weather (headwind and range) and planned limit of three hours flight time with four hours of fuel meant expected intermediate stops; as well as plan to descend and land if weather became too much. Flying due south from ZZZ2 IAS of 120kts; ground speed of 80kts dues to headwinds. Had picked up flight following from ZZZ Approach. Lowering clouds and rising terrain to the east (Mountains) led to risk assessment not to push minimums. Turned west (lighter cloud; and slower rising terrain) and decided to go VFR over the top. Proceeded southwest; maintaining ground contact. Moving to next clear view of ground it closed up.. first mistake.. should have turned 180 and used tailwind to get clear of cloud and land. Having found myself in IMC (no view of ground) and NOT turned 180; took the opportunity to descend when a hole presented itself. Second mistake.. hole too small to achieve 2000 feet separation so descended through IMC in a steep banked turn leading to rapid lost of height.. when clear through the cloud found the cloud base was lower than some of the terrain so still in IMC (NOT 500 below cloud). Spoke with center and asked for vectors to airport in VFR but was below their minimums. ZZZ4 was proposed and I started to head towards it; using the quadrant max height as a guide to safety. Assessment of cloud base and the position of the airport had me assess that it was too dangerous to get in there due to height of hills and not wanting to descend into a valley; or get squeezed into one. Knowing I had 2 hours of fuel onboard; and that I was in IMC although insight of the ground; and looking at quadrant heights; I assessed that it was safer to renter the clouds and seek sufficient ground clearance to avoid any terrain or structures. Should have [requested priority handling] at this point at the LATEST! Ascended into the clouds with concentration on the attitude indicator; making sure to make smallest corrections to keep wings level; and slow ascent so as not to get disoriented by large changes. Was fortunate that winds were constant and so not having to make any major corrections to able to keep things in tight control. I asked Center for another airport where VFR was in force and was given ZZZ5 which had (I think) 6000 overcast and plenty of visibility. Maintained altitude in cloud until cloud base lifted and was able to descend back to VFR; and approach ZZZ5 . Center stayed on after I reported airport in sight. Full pattern approach to ZZZ5 ; landed; and then called center on phone to report as unable to get a response on radio. Primary cause - flight into IMC due to under assessment of how quickly the clouds were closing up.. reality is that the gaps we were seeing were probably already too small for VFR and should have turned 180 probably 10+ minutes before. Second learning.. a turning dive into a too small gap over high terrain was a poor decision when the 180 was still an option - desire was to get back to VFR compliance ASAP when realized in IMC; and the best approach would have still been the 180 as we had separation from cloud layers and horizontal visibility to maintain see and avoid even though IMC; and with flight following. Third learning; should have [requested priority handling]; VFR pilot in IMC. I did not advise as my assessment was I had things under control and had contact with center. I was not worried about having to report in IMC; so likely over confident. Decision to go back into the clouds was based on belief that there was greater risk of CFIT (Controlled Flight Into Terrain) as still IMC; than to gain ground clearance. I should not have got myself in that position in the first place. I saw altitude and time as my best options; and to get back into VFR it was my fastest and safest path.