ZLC Controller reported issuance of IFR clearance to enroute aircraft while below MVA resulted in a CFTT event.

Narrative

Aircraft requested IFR pickup in the air. At first; he thought his original clearance was still valid but since he landed at ZZZ; his IFR was canceled. He was north of ZZZ when he requested an IFR so I issued him a clearance leaving 070 cleared to U56 via direct BYI direct PIH direct to U56 climb and maintain 090. Pilot read back the clearance correctly. Immediately after that I had to verbally handoff an aircraft that was failing to auto transfer to mountain home approach. It took a while to get the aircraft transferred and once the call was over; I looked back over to Aircraft X and saw that he flew south into higher terrain. I quickly asked if he was direct BYI (which is what I expected) and he confirmed that he was; I issued a low altitude alert; waited until he reached 070 and after confirming he was in the 070 box (zoomed in) I turned him left 20 degrees to keep him clear of terrain. It wasn't until after I reported the event; that the falcon playback showed the aircraft was very clearly in the 090 box when I issued the turn towards lower terrain. I wish it was easier to identify how close the aircraft is to terrain. The difference between the radar scope and the falcon surprised me. Next time; I will just suggest a heading away from terrain. Honestly; since the pilot did do an unexpected turn to the south; my first thought was that he needed course guidance to help him. I also would have noticed the aircraft flying south sooner if my attention was not taken away by the failed auto handoff.