Arthur Davison appeared as a guest for Exeter City during 1945/46 when two interim league competitions - Division Three South (South of Thames) [played until Christmas] and the Division Three South Cup (Southern Section) [played from January] - were organised ahead of a return to ‘normal’ league football in 1946/47.Davison guested for Exeter City during the 1945-46 season. He played one game for the Grecians in a 5-1 defeat at Cardiff City in February 1946, in a Division Three South (Southern Section) cup-tie.
He had signed for Stockport County in Septemebr 1945, making his debut for them on the 15th of the same month against Crewe Alexandra. He played a total of 6 games for Stockport County before leaving in November 1945. Davison then moved to Torquay United in November 1946 and appeared in one Division Three South fixture for them, a match on Christmas Day 1946. Torquay lost 6-1 to Crystal Palace at Selhurst Park in front of 11 786 spectators. He also played at full back for Torquay United Reserves, and was still at Torquay in November 1947.