Sam Allardyce's team rose briefly to the top half of the table after a 2-0 victory over Norwich City on Tuesday, only to be overtaken by Swansea City 24 hours later.
Still, the improvement in league form and position since the turn of the year is evident, with West Ham having sat 19th on New Year's Day with only three top-flight wins all season.
The gap to Sunderland in the bottom three is now four points and West Ham are unbeaten in four games, but Collins will not be relaxing until they have reached 40 points.
"The pressure is never off until you've got the sacred 40 points," he told the club's official website. "We knew that, on Tuesday, that we'd gone into it with three good results and it would have been quite easy to think we'd done the job.
"But it was a massive game, we realised that, and to get the three points made it a big night for us."
With West Ham out of the FA Cup, they can enjoy a break from action until the visit of Southampton on February 22, and Collins believes that rest will aid their cause.
"The break between fixtures is helpful, there did seem to be a few tired legs out there late on Tuesday," he added. "It's been tiring on the brain too, because with the position we're in you're always thinking about it.
"That takes its toll, and winning three in a row will lift a weight off a lot of people's shoulders and allow us to go out there and enjoy our football again.
"When you're down there you wake up in the morning and that's all you're thinking about, so to lift that is a massive boost."
