ZME Controller described an event requiring multiple vectors to avoid a military restricted area; noting changes in the current Letter Of Agreement should eliminate these type delays.

Narrative

Air Carrier X departed Birmingham (BHM); left on a heading of approximately 330 degrees assigned by BHM Approach Control. Atlanta Center; Birmingham Sector (ZTL 14) left aircraft on heading. I attempted to contact ZTL 14 to have them turn Air Carrier X on course to avoid the Columbus 4 MOA which was active 8;000 to FL210. I called on the 'shout line' 4 times and got no response. I accepted the hand off on Air Carrier X after trying to get hold of ZTL 14 with no success in contacting them. When Air Carrier X checked on my frequency; the pilot advised me he was on an assigned heading. I advised the pilot; due to ZTL 14 not answering the land line; I was unable to turn him on course for approximately 40 miles in order to miss military airspace. I did give the Pilot the option to make numerous turns to the southwest in order to fly his original route; but he said he was willing to stay on his present heading. In the vicinity of the Muscle Shoals VOR (MSL); I was able to turn the aircraft westbound towards Memphis; TN. Recommendation; The Letter of Agreement (LOA); changes between Memphis Center and Atlanta Center; in regards to Birmingham departures; needs to be amended to have ZTL; Birmingham Sector put BHM departures entering ZME Hamilton Sector on course. To delay Air Carrier X as long as I was forced to due to a poorly written LOA and the laziness of the Controller at Birmingham Sector to answer the 'shout line' is inexcusable. If the Letters of Agreement were changed to reflect that aircraft will be cleared on course by ZTL Birmingham Sector should eliminate these issues.