A Center Controller reported an aircraft deviated from its assigned route and flew below the Minimum IFR Altitude.

Narrative

Flight data calls my sector requesting an IFR departure out of ZZZ to ZZZ1. I clear the aircraft short to ZZZZZ via the ZZZZZ Obstacle Departure Procedure. Aircraft gets off the ground and is approaching ZZZZZ when they call and ask for further clearance. I double check their flight plan; and they are filed for something completely wrong so I issue the pilot 'after ZZZZZ direct ZZZ2 then direct ZZZ1; and I will get the rest of the route between ZZZ2 and ZZZ1 in a moment. I didn't have time to find the official routing at the moment due to traffic complexity and volume; but I knew that the routing I gave would be safe until I could get back to the pilot for the rest of the route.The pilot then turns direct ZZZ1 which puts them below the MIA so I come back and tell them 'Aircraft X low altitude alert; check your altitude immediately; the MIA in your area is 12000.' The pilot then turns back to ZZZZZ and another controller plugs in on my D side and coordinates with the next sector a heading to put the aircraft on and they will finish off the flight plan. Don't really have any recommendations; it all just happens very quickly in that area due to airspace size and configuration.