The Hammers were in deep trouble just a few weeks ago but a run of four straight wins has catapulted them into the top half of the table.
However, just seven points still separate Sam Allardyce's men from the bottom three and Cole insists there is no chance of thinking safety is already guaranteed.
The striker is staying focused on Saturday's clash with Everton and hopes West Ham can continue to climb the standings.
"We're not safe yet, we just need to keep winning games," Cole told the club's official website. "We are looking upwards. At the start of the season, we were looking up too and it is good to do that rather than looking over your shoulder, because that can lead to mistakes.
"We now have to look towards Everton, which will be a hard game.
"They are a good footballing team so we just need to nullify their attacking and try and get a goal, because the form we are in at the moment, we can notch against anyone."
Cole netted his sixth goal of the season in the 3-1 victory over Southampton and is delighted with his form, having returned to the club on an 18-month deal last year.
"I didn't have a pre-season, but I have been playing and that is what I need to do," he said.
"Training has been intense. There is a lot of work that goes on behind the scenes to keep me fit. I feel wanted because I got re-signed.
"When you are not feeling wanted, it is always hard to take, but when you feel wanted it is a big difference to your game and your attitude."