PA-28 Cherokee/Archer/Dakota/Pillan/Warrior
Air Traffic Control Fully Certified
Air Traffic Control Time Certified In Pos 1 (yrs) 1.5
Situational Awareness; Confusion
ATC Issue All Types; Deviation / Discrepancy - Procedural Clearance; Inflight Event / Encounter CFTT / CFIT
Human Factors; Procedure; Chart Or Publication; Airspace Structure
Aircraft X was doing a practice approach. Upon conclusion of his approach; I issued him direct leaving 3;700 feet which was the MIA(Minimum IFR Altitude) to maintain 4;000 feet. After communications transfer to TRACON there was a MIA of 4;100 feet that Aircraft X clipped. The portion of flight through the elevated MIA was about 3-5 miles. I didn't call TRACON at that point because the aircraft was going to exit the higher MIA just as fast by remaining on course as a vector would and there was IFR traffic at 5;000 feet.  I didn't realize the MIA change until the MSAW (Minimum Safe Altitude Warning) went off. Maybe have an earlier MSAW alert before the aircraft penetrates it but it was my mistake.
A Center Controller reported they issued an aircraft a direct routing and did not realize the route was below the Minimum IFR Altitude.