Graydon, who can play at full-back or wing, is brother of Wallaby winger Scott Staniforth, has represented Australia Sevens on the IRB circuit.
His senior club career started in 1999 with Australian Universities and Southern Districts before moving onto the New South Wales Waratahs Academy and B side as well as representing Australian Legends in the Hong Kong 10s.
With a reputation for powerful, pacey running and a long accurate kicking game, he moved to play for New Zealand NPC team Southland in 2003 before moving to Exeter Chiefs last season. He was instrumental in Exeter's drive for promotion to the Guinness Premiership but they were edged out by Bristol.
As his wife is a UK passport holder, Graydon will qualify as a European player when playing for Glasgow.