The 30-year-old was released by Sam Allardyce's side in the summer but due to an injury to Andy Carroll and a lack of forwards at the club, the former England international was re-signed on a free transfer.
"I want to get back into the thick of things now," Cole told the club's official website. "I had a taste of the bench on Saturday against Manchester City and I just wanted to get on so much.
"The manager felt it was best not to throw me in at the deep end so fast, to try and get this game [West Ham's Under-21s' 2-0 victory over Wolves] under my belt and be available for selection in the other first-team games.
"Hopefully I can convince the manager to get me in the squad, coming on as a sub or even starting a game sometime soon."
West Ham are 15th in the Premier League after eight games this season.