The attacking midfielder netted for the first time since returning to Upton Park as the Hammers were held to a 1-1 draw by London rivals QPR on Saturday afternoon
The 31-year-old ended a frustrating three-year spell at Liverpool with a move to his boyhood club earlier this month.
And, having scored a close-range equaliser as the Hammers were held to a draw on Saturday, the former Chelsea man is keen to keep contributing.
"It was nice to score. I know we're going to put a lot of crosses in and it's up to me to try to get on the end of things," he told the club's official website.
"It was good and, of course, the main thing was to get a point and keep us moving forward and getting points on the board.
"It's always nice to score any goal. When you have days like that in football, you just have to try and enjoy them. I'm happy and playing with a smile on my face.
"Playing in that position a bit further forward out wide, you have got to score goals. Myself and Matt Jarvis are working on getting in the box and scoring more goals.
"I thought it was a good, strong team performance and we bounced back from a poor performance last week [at Sunderland]."
Despite being happy with his goal and the overall team performance, Cole expressed his belief that the hosts had deserved more than a point.
"It was a game we should have won. We had a lot of possession and put a lot of crosses in the box but credit to QPR because they defended strongly," he said.
"It's two points dropped, but we'll try to take the positives. It's another draw and a point on the board and we'll go to Arsenal on Wednesday in full spirits knowing that we put in a good performance on Saturday.
"I think 38 points will probably be enough this year [to stay up]. Like I said, you can only perform well and sometimes things don't fall for you - you have balls coming off people's backsides and that.
"It was a good performance I thought from the team, strong and together. It was good to have Joey O'Brien back from injury as well, and Mo Diame, of course. We looked strong."
The Paddington-born midfielder has been beset by injuries in recent years, but feels his physical condition is constantly improving.
"I felt a bit stronger. It's just about getting my body hardened to playing 90 minutes again. It's nice to have an effect on games. I have played three games now and I've had an effect on them," he added.
"It's good and I'm enjoying it - I'm enjoying my football and that's the main thing. I'm just settling in and all I want to do is be part of a good team and a good bunch of lads, help my team to win and enjoy my football - that's all I want."
"We'll go to Arsenal to win the game. We have done it before as West Ham. We've just got to go there and frustrate them because they have got great footballers - your Jack Wilsheres, your Theo Walcotts and people like that.
"We'll go there and try to stifle them a little bit and we'll always be a threat from set pieces, so we'll go and have a go.
"If we keep performing like we did on Saturday, then I think we'll win games."