Center Controller reported they took over a sector with an aircraft entering an area below the minimum IFR altitude.

Narrative

I was relieving the controller on sector just prior to the event. The controller being relieved gave me the position relief briefing for the sector and I acknowledged that I had the sector by giving my initials on the recorded line. The controller being relieved unplugged his headset; moved his break tile to the 'available' side and left the control room all within one minute of me taking over the sector. After I sat down; I noticed Aircraft X who was at 6;000 ft. and had 'MSAW' flashing in their full data block. I immediately displayed the MIA (Minimum IFR Altitude) map on the ERAM (En Route Automation Modernization) scope and realized that this aircraft was about one flying minute from entering a 6100 ft. MIA. At the time of the position relief briefing; this aircraft was in the process of handing off to ZZZ Approach and no mention of the approaching rising MIA was mentioned by the controller being relieved. I tried to contact Aircraft X numerous times on the main; standby; and Backup Emergency Communications (BUEC) transmitter with no response. Due to the location of the aircraft; it was unknown to myself which frequency they were on. There was one aircraft; I believe the call sign was Aircraft Y; that was on XXX.X at 8500 ft. I asked this aircraft to relay to Aircraft X to see if they could make contact. Aircraft Y replied back to me saying they could hear each other. I told Aircraft Y to tell Aircraft X to climb to 7000 ft. immediately for the MIA. Aircraft Y transmitted this to Aircraft X and said that they acknowledged the transmission. I then observed Aircraft X being a climb and eventually level at 7000 ft. During this time I also called ZZZ Approach to notify them of the situation and asked them to relay on 121.5 for this aircraft. ZZZ Approach also requested permission to broadcast on XXX.X from their own transmitter to attempt to reach this aircraft; which I allowed. The Area Operations Supervisor was notified immediately of the situation. First and foremost; more attention being paid by the controller being relieved and more attention being paid by myself during the briefing could have fixed this event sooner. However; once I realized the developing situation; I had no way to communicate directly to Aircraft X. This aircraft was above the MIA at the time and inside my sector boundary; but I had no way to contact them directly. This is an issue. There have been reports filed in the past about XXX.X and it's limitations. Aircraft X was on XXX.X and I could not talk to them. Something needs to be done about this sector and its frequencies. Aircraft X was in an imminent situation that could not be fixed due to the poor radio coverage. Due to this; it puts the controller at the sector at a disadvantage due to them being unable to fix the conflict.