A Center Controller reported a departing aircraft was issued an incorrect departure procedure and flew below the Minimum IFR Altitude.

Narrative

Aircraft departed LWS airport and was not radar identified due to typical low coverage area. After a few minutes; requested a direct short cut. Was direct POTOR in the Flight plan. Upon radar identifying the aircraft; I noticed he wasn't flying the usual route to the ODP (Obstacle Departure Procedure) which goes to the MQG VOR or turning to the next fix on his flight plan. I immediately climbed the aircraft to the 2nd next highest MIA (Minimum IFR Altitude); since the first MIA would be quickly eclipsed by the higher one. 7000ft. Upon questioning the pilot; he stated he was given runway heading up to 6000 ft by the tower. This is not permissible at LWS. We have no LOA and no legal way of separating from terrain. After issuing the MIA to the pilot and again and stating they were headed for higher terrain; I waited until the aircraft was safely above the MIA and gave them a shortcut. My D-side called tower and to ask if they issued Runway heading; and they said no. I recommend that there be a clarification on the departure instructions from the tower. There has been more than one instance where an aircraft has departed runway heading.