The Hammers only face one match against a team that finished in the top six of the Premier League last season until November when they host Arsenal in October, and Allardyce is hoping that his side can get points on the board before they face sterner tests during the winter.
Allardyce told The Mirror: "Our fixture list has not given us any of the big boys early on so you have got to try and do as much as you can and take advantage of that.
"No game is easy as a newly-promoted team, but it gives you the chance to play your best and secure some points.
"These fixtures are critical for us until November when the big games start flowing each week."
West Ham began their Premier League campaign with a 1-0 win over Aston Villa at the weekend.