A win at home against already-relegated Reading on Sunday will confirm a top-half finish having only been promoted through the Championship play-offs last season.
And Allardyce believes this is an excellent platform to build from for next season and beyond.
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"We split the season up into phases and we thought that our maximum ask this year, if everything went according to plan, would be tenth in the Barclays Premier League.
"We have tried everything to reach it and aimed for that. If you say 'Come on, we can finish sixth', they just know that's going to be a tall order.
"Tenth was our maximum which means I think we've over-achieved this year."
Allardyce believes his previous Premier League experience has helped West Ham this season, though he admits his team's recent away form has disappointed him.
"Having been in the Premier League for as long as I have with three clubs then you know what to expect," he added.
"You know when the pitfalls are going to appear and how to overcome those pitfalls and you strive to make sure that the players stay in a very good, positive frame of mind. That's when you do well.
"Calling on all that experience has, I hope, helped them this season and of course the response I've had from all the players, staff and fans has been very, very good.
"We've had a consistent, well-constructed season apart from our away results where we've not scored enough goals or got enough results. That's the only one disappointing thing."
Allardyce also revealed the club want to sign Andy Carroll as quickly as possible after the striker was recalled to the England squad on Friday.
"[Andy Carroll] has shown everybody what he can do and he has massive potential to grow into a very good player if he ends up back here with us," he continued.
"It's about negotiation and both parties have to be happy that the negotiation cannot go any farther and it's a 'Yes' or 'No'. We hope it's a 'Yes' and we hope it is as quickly as possible."