The former Bolton and Newcastle boss has already brought in four players this summer but is still on the market for a player who can "come in and make a real difference"
The Hammers have already secured Andy Carroll on a permanent deal from Liverpool, Razvan Rat from Shakhtar Donetsk, Adrian from Betis and teenage midfielder Danny Whitehead from Stockport County this summer but the 58-year-old wants another "top-quality" player who could help them to improve on last season's 10th-placed finish.
"What I want in that one player is top quality, I want that player coming in to challenge the players already here for their place immediately," Allardyce told the club's official website.
"I want somebody to come in and make a real difference, as we did last year when the players we brought in made a massive difference. That took us to the next level and led to us having a fantastic season and finishing 10th."
Allardyce concedes that his overhaul of West Ham's squad in their one-off campaign in the Championship in 2011-12 was a big burden for the London club but he is thrilled to now have some stability.
Only six players have left West Ham since last season, with Carlton Cole the only regular performer in the senior team to depart, and the manager is convinced that will help his team in his third campaign at Upton Park.
"For me, it's two years in and fewer players will have to leave or come in to the club, which is great for me," Allardyce remarked. "The last two seasons have been traumatic in change-around terms, particularly the first one when I look back on it.
"It was horrendous, with something like 28 players going out and 20 coming in during one season. We had to cope with that and rebuild the football team.
"At this moment in time, I'm a lot more comfortable because the players and staff know what I am looking for and how I work. They know what I want and it is their responsibility to drive the players forward in every department to extract even more out of them this year than they did last season."