The 31-year-old midfielder returned to his boyhood club in the January transfer window after he failed to nail down a place in the Liverpool side.
He progressed through the ranks at Upton Park but only ever tasted success in the Intertoto Cup.
Now Cole, who played the second half of West Ham's 3-0 win against Boreham Wood on Wednesday night, wants to see the club improve on last season's top-10 finish and collect a winners' medal along the way.
"I think the ambition of the club at the moment is improvement," he said.
"But for me personally, I would love to finish better than 10th but I would love to take this club to Wembley because at the end of the day they are the days you remember in your career - it would be a massive achievement.
"You look at Swansea and Wigan last year and that is their day, we would like a bit of that because it has been such a long time since West Ham have been there, so that would be nice."
Sam Allardyce led the Hammers to 10th on their return to the Barclays Premier League and has already added to his squad ahead of the new campaign.
The stand-out signings are those of Andy Carroll, who spent last season on loan at West Ham from Liverpool, and Romania captain Razvan Rat - brought in on a free transfer from Shakhtar Donetsk.
Cole believes the calibre of such players proves that West Ham are able to attract big names.
"It is a boost for the club and shows where we are going," he added.
"Andy Carroll is a young, England centre-forward and clubs do not usually sell them and he is a big talent. He did a good job and we just want him to get fit and come back because he is a big part of our team.
"West Ham is a big club but it has been starved of success. You look at where we are in London and people want to come here and play in London and play for West Ham.
"It is great that players like Razvan and Andy are coming in and hopefully there will be a few more as well."
The former Chelsea ace has had his fair share of injury problems in recent years and admits his first aim for the forthcoming season is to play as often as possible and even maybe attract the attention of England manager Roy Hodgson.
"I always keep my goals to myself and take it day-by-day but I just want to keep fit first and foremost," said Cole.
When I was fit last season my performances were good so the key for me is just staying fit and if I do that, the football for me has never been a problem.
The ambition will always burn every time you go out on the football pitch and you want to do well, if you're an Englishman and you're playing well you are inevitably going to be talked about.
I have played over 50 times for my country so if I can get playing you never know but at the moment I'm just concentrating on West Ham.
The 3-0 win at Meadow Park gave Allardyce another chance to look at new signings Adrian, Rat and Danny Whitehead, with the Premier League outfit fielding a strong side in either half.
Ravel Morrison tapped home on 20 minutes after Boreham Wood goalkeeper James Russell had made a sprawling save to prevent Rat's deflected cross creeping in.
The second came just moments later as Ricardo Vaz Te headed home after a short-corner routine but, despite the visitors dominating for large periods, they did not add to their lead until skipper Kevin Nolan volleyed home with just under 10 minutes remaining.