ZBW Controller reported limited radio capability to issue a climb clearance resulted in an aircraft entering a higher MVA and a CFTT event.

Narrative

Aircraft X from WVL-ZZZ was coming to us from Bangor approach. They flashed them early and told us Aircraft X was able 070 and our control if we needed it. About a half hour earlier we discovered the BGR site for frequency XXX.XX was broken so we told BGR to use YYY.YY which is also located at BGR. We waited for the aircraft to come over so we could climb them to avoid a 058 MIA that is just outside of BGR approaches airspace. BGR called and said Aircraft X tried YYY.YY and could [not] reach us. We told them to try it again and if no luck to use XXX.XX as the site was working so we eventually would get the aircraft it just may be a bit before we can talk to them. At the time Aircraft X was about 2 minutes from the 058 MIA while still at 050. We called BGR back and asked them to climb Aircraft X to 070 which they did. By the time Aircraft X was climbing they hit the 058 shelf while starting the climb out of 050. We were still unable to communicate with the aircraft and once they were clear of the shelf and above 060 we finally got communications with them on the site.Suggestion: Even though we didn't know about YYY.YY being down at the time we should have told BGR to climb Aircraft X when they first called to prevent any more frequency issues that could happen and we eventually did run into. Starting off by using positive control and getting the aircraft climbing where we know they are communicating with someone to ensure a safe altitude would have prevent them being below the MIA.