A third win in four games was secured on Sunday thanks to Ricardo Gardner's superb winner against Newcastle at the Reebok Stadium.
It means Bolton go to Stoke on Wednesday with renewed optimism, given their recent form.
Steinsson said: "We are seven points away from the relegation zone with 11 games left and have to be looking up.
"We are doing well and the challenge is to close in on teams like West Ham and Wigan.
"I have never been in a club where there has been such a good atmosphere. It is really great to be part of things.
"I think the fans can see the players wear their hearts on their sleeves. We are giving it everything we have and you can see that in our results. We are getting there slowly."
Manager Gary Megson is not a stickler for routine either when it comes to day-to-day work.
In the build-up to the Newcastle match he took the players into the Cheshire countryside for a paintball session.
"We had a great time," said the Bolton boss.
"I like to see them doing some activity that is not football-related. They all seemed to enjoy it.
"You would not one trust one or two of them with their shooting and they are even worse with a paintball gun!
"The players have also gone go-karting this season and we visited Dubai as well. You need to let them relax a little bit as there are a lot of expectations on them when they play."
Meanwhile Steinsson recognises Stoke are doughty opponents, especially at the Britannia Stadium, and will be fired up after taking a point from Aston Villa.
"They are a big team with strong players and this is another massive match, " the former AZ Alkmaar full back told the club's official website.
"It will be a hard, physical game."
Stoke 11/8, Bolton 2/1, Draw 9/4