The 34-year-old, who joined the club from Burnley in 2011, scored nine goals in 47 games for the Blues last season.
"I'm delighted we have got Wade back. He was a really important member of the team last season. He worked very hard for the team and scored some valuable goals," Clark told the club's official website.
"He is fantastic example to the younger players and I really wanted him to be part of this season so it's great to have him back."
Elliott began his youth career at Southampton.