Mark Noble's penalty and a James Chester own-goal were enough to beat 10-man Hull at Upton park on Wednesday, though the visitors dominated much of the game and had chances to secure a point.
Jeers rolled down from the stands at the full-time whistle with Allardyce cupping his ear in the direction of the disgruntled supporters and criticising them afterwards.
And with a trip to Sunderland to come on Monday, Allardyce - who has endured a prickly relationship with the West Ham fans since taking over - believes the team's current position of 11th may even be better than they should be aiming for.
He said: "I thought we overachieved last year finishing 10th. No doubt about that.
"We had a fantastic season. Over-achievement always brings a greater expectation. That's why we have suffered at times this season, particularly at home.
"Last year we were immense. We only lost four games at home last year in the entire season. That was to Arsenal, Tottenham, Everton and Liverpool.
"This year at home has been much tougher. (But) we have put it right recently with four home wins from the last five games.
"We should finish no lower than 12th or 13th. That's about our top level.
"It would be the same as last season. We are building steadily and slowly."
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