ZLA ARTCC Controller reported that a known radio coverage problem over an area led to a MVA violation.

Narrative

Aircraft X was an IFP (Instrument Flight Procedure) pickup off of HII. He had called me multiple times trying to receive his clearance. However; because Sector 6 has two radio transmitters; he continually was stepped on by other aircraft on frequency. There was a lot of frequency congestion; and when there was a break in that; I reached out to Aircraft X because I could hear his partial call sign the other times he tried reaching me. Once I acknowledged him; he said he was looking for his IFR clearance to ZZZ1. The flight plan that was listed in my system showed he had filed his route direct ZZZ1 at 070. He was right on the border of Sector 6 and Sector 10 at the time I DM'ed (Departure Message) his flight plan. Once I tracked him up; the data tag immediately started flashing to Sector 10 as a hand-off. I called Sector 10; and requested control to make him IFR. Sector 10 took a point out on the aircraft. I cleared him direct to ZZZ1 at 070; as was listed in the flight plan.The MIA (Minimum IFR Altitude) blocks in the near vicinity were all below 070. There was a 074 MIA block approximately 20 minutes away from his location when I cleared him. I could not get an accurate route line to display on my scope because of preferential routing listed in the computer. However; it was not initiating it for me to issue to the aircraft. As I was attempting to figure out this situation; other aircraft were calling me and I had other things I needed to figure out in my sector. The aircraft was flashing to Albuquerque Center Sector 43. Approximately 1.5 minutes outside of their boundary; A43 called me; telling me to climb Aircraft X to 090 because of poor radio coverage in the area. I confirmed with them that if I climbed the aircraft to 090 they would be able to hear him; and they said affirmative. I got off the line; and immediately attempted to climb Aircraft X to 090 but I received no response. I switched my QQF radio between mains and BUECs (Backup Emergency Communications System) to attempt to reach the aircraft on both and I asked another VFR aircraft in the vicinity to relay. Aircraft X was already in A43's airspace. There was no response. I notified the CIC and A43 that the aircraft was NORDO. I then attempted very frequently to reach the aircraft on the QQF mains and BUECs; and on Guard as well. In between calls to the aircraft; I was making transmissions to the other aircraft on frequency. Aircraft X was approaching the 074 block. A43 had one of their VFR aircraft change to my frequency and attempt the pilot; and that pilot told me that Aircraft X received the message to climb to 090. Aircraft X entered the 074 block at 072; but did climb above the 078 block that he entered next.I am aware of the poor radio coverage in that area; and I should have climbed the aircraft to 090 upon issuing his IFR clearance. And; since by the time I was able to reach out to the pilot; the aircraft was only about a minute from Sector 10's boundary. I should have instructed him to make his request with Sector 10 or A43; whose airspace he entered shortly thereafter. I also could have just radar-identified him and instructed him to maintain VFR; and coordinated with A43 with his altitude and route clearance. The radio coverage in Sector 6 is spotty especially in the southeastern corner of the Sector 6; which many times leads to increased workload. Our staffing was very low on this day; and I had two 20-minute breaks and two 30-minute breaks during the day. This was my last session of the day. Sectors 6 and 16 had been combined up most of the day; and I had opened Sector 6 up during this session. When I opened up Sector 6; I was busy right from the start; and leading to this event.