The 30-year-old has penned an 18-month contract with the Hammers after returning to the club on a short-term agreement in October, having originally been released last summer.
With West Ham struggling for goals and with record signing Andy Carroll still sidelined, manager Sam Allardyce offered Cole a three-month deal to help get the club firing again and he has done enough to earn a longer extension.
The England international has scored four goals in 11 Barclays Premier League fixtures since his return, including the opener in the vital 2-0 victory at Cardiff on Saturday that took the Hammers out of the relegation zone and now Cole hopes to reward Allardyce's renewed faith in him by continuing to find the back of the net.
"I'm really happy," he told West Ham TV.
"I've worked hard to get to the position I'm in right now. The manager asked me to come back in the summer and I've worked hard to get my match fitness and I've scored a few goals whilst doing that.
"I've always believed that I can score goals and that's the main thing as a striker, you want to score goals and help the team win. That's what I'm here to do and hopefully I'll get more chances to do that.
"I've never said that I've got targets but I have my own targets in my head. Hopefully I can get to where I want to get to. First and foremost it's all about Premier League survival, we're down there fighting it out and we just need to stay up and then work from there next season."
Despite offering Cole a new deal, Allardyce is likely to continue his pursuit of Monaco forward Lacina Traore, with a move for Inter Milan's Ishak Belfodil seemingly dead in the water.
Saturday's win at Cardiff was West Ham's first in the Premier League since November and, with Allardyce welcoming former club Newcastle to Upton Park this weekend, the Hammers need to continue to pick up points if they are to avoid dropping back into the bottom three.