The 30-year-old's seven-year association with the club appeared to have ended in the close-season when he was released upon the expiry of his contract.
However, a heel injury to Andy Carroll left West Ham with a shortage of forwards, prompting manager Sam Allardyce to offer Cole a short-term deal in October.
Cole went some way to repaying that decision when scoring the second in West Ham's 3-0 triumph over Fulham on Saturday, a result which ended a five-match winless Premier League run and ultimately cost Martin Jol his job as manager at Craven Cottage.
The former Chelsea and Aston Villa man is now hopeful of staying with West Ham beyond January, when his current deal is set to end.
"Once you score you feel you can get in a rhythm and score more so hopefully there will be more to come for me now," Cole commented.
"Hopefully they will be important goals. I want to keep the momentum and doing the right thing.
"I want to stay, of course, and I'm very confident. I know the ins and outs of the club. Sometimes change is good but I'm here and have been here for so long - I'd like to stay on.
"It's up to the manager. I can only play my best and try to play to my capacity on the field. Whatever they will do in the future is beyond my control. We'll see how it goes."
Cole has yet to begin a match since returning to the club - making four appearances from the substitutes bench - and he is targeting a starting role in the near future.
He added: "I'm fit now, I've always been working for my fitness and the manager has been waiting for the right time for me to start.
"Hopefully I've done myself a favour and will get a start soon. I'll wait for the manager to make the decision."
