ZLC Center trainee reported they gave approval to an adjacent sector for an aircraft to enter their sector below the Minimum IFR Altitude.

Narrative

I was getting briefed to take over sector 17/66 for my first session of the day. The controller briefing me was getting a landline call from ZMP but didn't answer since he was giving a position relief briefing. Another controller answered the call for us. When I finally took over the sector; the controller that answered the landline briefly mentioned that the call was for an IFR nonradar handoff from ZMP 23 (south) and the aircraft would contact me at the boundary. I was slightly busy with cleaning up the sector when he told me this; so I thought everything with that aircraft (Aircraft X I believe) was taken care of. I coast tracked the data block so it would follow his route and moved on. I kept coming back to see if the aircraft had come back on radar and eventually it did; so I tracked it to the target and immediately the Minimum Safe Altitude Warning (MSAW) alert started flashing on his data block (FDB). I realized he had entered a 065 MIA box at 060 feet. I tried to tell him to climb to 065 but no response as radios are bad in that area; especially that low. I got a relay from an Aircraft Y who successfully told Aircraft X to climb to 065. I then gave Aircraft X a low altitude alert and told him to report reaching 065. He heard this transmission and a minute or two later reported level at 065.It has always been a habit for me to double check routes/altitudes of aircraft at lower altitudes to avoid this exact situation. Unfortunately today I had a lapse in judgement and trusted that things were taken care of when I took over the sector; since it's also a habit of mine to leave the relieving controller with a mostly clean sector and no conflictions to handle. I should have double checked Aircraft X and called ZMP back immediately to climb the aircraft at least to the MIA. When the relieving controller wasn't answering the call I should have made it a point to go back to why ZMP was calling in the first place. I could have called them back myself and told them that someone else had answered the call and to repeat what it was about. I know for a fact that if I had called them back myself I would have checked the aircraft's route/altitude and seen the issue; as that is muscle memory for me when answering landline calls especially regarding nonradar aircraft. I think I did everything I could when I finally realized the problem; I just should have obviously caught it sooner.From now on if anyone else answers a call meant for me I'm just going to call back and have them repeat it. Playing telephone with other controllers in such a serious career field is obviously ineffective and can cause major issues. I also need to remember to always double check mine and other controllers work because everyone misses things sometimes. I will put even more attention to aircraft routes/altitudes to make sure they meet the MIAs and prevent this from happening again.