Cole has won trophies in his career during his time at Chelsea but is still hungry for success and admits it would particularly special to get a medal with the Hammers.
The former England midfielder came up through the ranks at West Ham and returned to the club in January after two-and-a-half years at Liverpool.
"There will always be the street footballer inside of me," he said. "But you want to do something more than that in the game.
"You want substance and impact. I got that at Chelsea. That is why it would be so special to win some silverware with West Ham.
"I'm lucky enough to know what it feels like. And I saw the fans at the 2006 FA Cup final, when they came so close.
"To win something with a group of supporters who keep the faith like that even though they have been starved of success would be so amazing.
"I hope I can give the fans some big moments here yet."
Cole will come up against his former Chelsea manager Jose Mourinho when West Ham host the Blues on Saturday.
He appreciates that he owes a lot to the Portuguese, but has also paid tribute to current boss Allardyce for the way he has transformed West Ham's fortunes over the past couple of years.
"I've got a lot of time for Sam," said Cole. "He has turned West Ham around. Although you can't really compare West Ham and Chelsea and the money, the job that Sam has done is just as remarkable as Jose did at Chelsea.
"He knew what to say to players at the right time. It wasn't always what I wanted to hear, but it was what I needed to hear. I was a decent player before I went there. He definitely made me better, whatever the perception people have.
"He got my mentality right. He saw something in me that maybe I hadn't. He changed me from playing central midfield to playing wide and playing a bit higher up the pitch. I became more effective at both ends of the pitch.
"I can see his impact already now he is back at Chelsea. With him they are a better side already than last season. They look more solid, more together."
Cole is confident that West Ham can push on and improve on last season's tenth-placed finish, whilst also challenging in the cups.
He said: "I'm still hungry for more. At West Ham we want to be finishing like we did last year and have a go in the cups.
"In many ways it is still the same club, but now there is a feeling something is building. The new stadium gives us a chance to grow because it will get harder and harder to keep up with the big boys.
"It's been a struggle at home this season, especially after we started so well. But we've just got find a way to get goals." 'I hope I can give the fans big moments'."