The Hammers showed all their fighting qualities to get their season back on track last month with an impressive draw against Liverpool and then consecutive victories over Burnley, Hull and Swansea.
Following the disappointing Emirates FA Cup defeat against Manchester City in the last game, Bilic is now looking for a similar response when his team play a big London derby at London Stadium on Saturday.
Bilic knows it is not going to be easy against a Crystal Palace team who are also fighting to move away from the bottom three and says his team will have to be at the top of their game if they are to come away with the right result.
Bilic said: “We have a gap between us and the relegation zone, but we have to continue picking up points and be as safe as soon as possible. Everything will then look brighter.
“We have a chance on Saturday, a big game at home to a club below us and we want to win and be closer to mid-table.
“They have a good team and like us they are not in the position they planned or thought they would be. It happened to them last season when they had some ups and downs. They started the new year fifth in the league and now they are starting another year fourth from bottom.
We have a gap between us and the relegation zone, but we have to continue picking up points and be as safe as soon as possible. Everything will then look brighter.
Slaven Bilic
“They have a very good team, a good balanced team, but they are not picking up points even though they played well against Watford and should have won.
“They lost against Arsenal which you can understand, but the last game against Swansea was probably a big shock for them. They are without a few players through injury and Zaha at the Africa Cup of Nations.
“They have players back in training, McArthur, Scott Dann, maybe Benteke. It's going to be a demanding game in every sense. It is also a nervous game for both teams, because of what's at stake.
“It's a great opportunity for us to get back to the form we showed in the league in the last five or six games.”