Controller reported an aircraft was on incorrect heading and not climbing to required altitude; resulting in a MIA deviation.

Narrative

I had just received a position relief briefing from the Controller being relieved and was told that Aircraft X was IFR and was going to be executing the published missed approach to KREEN but will want their IFR clearance to JRF. Shortly after taking the position; Aircraft X checked in stating that they were climbing VFR and wanting to pick up their IFR to JRF at 7;000 ft. I then asked the pilot to verify that they were canceling IFR and going to proceed VFR at this time. The pilot responded that they wanted IFR at this time. I then asked the pilot if they were on the published missed approach and the pilot advised that they were flying runway heading at this time. I advised the pilot that you understood that he was assigned the published missed approach procedure to KREEN. The pilot advised they were not issued anything and was just told to contact you on departure. I then pulled up the EOVM map and vectored Aircraft X to a heading of 055 for terrain. Then I went back and told Aircraft X to climb and maintain 4;000 ft. since the missed approach only would climb to 3;000 ft. and I knew I would need Aircraft X at 4;000 ft. to the east of LIH for his IFR. Aircraft X was apparently in the 7;000 ft. MIA when I issued climb and maintain 4;000 ft. Aircraft X was only out of 2;400 ft. and climbing when he reached the 4;000 ft. MIA 50 seconds later. Eventually; I issued Aircraft X their IFR clearance back to JRF once they were mid channel. I don't have any recommendations for this instance.