Everton beat Swansea City 3-1 on Monday and pushed themselves up in the ninth in the Premier League.
“The target now? Top half, but let’s not get carried away,” said Allardyce.
Everton were in the relegation zone when Ronald Koeman was sacked after a humiliating 5-2 loss at home to Arsenal.
He was followed by caretaker David Unsworth who managed two wins in eight games. His second and final win was a 4-0 against West Ham with Allardyce already watching from the stands.
Although he was on the bench, it looks like he was making sure that his team would win and that the victory counts on his record.
“To collect 13 out of 15 points is outstanding considering the position where we have come from. We have scored 11 goals in five games and kept clean sheets. We are a team that comes back from 1-0 down to win and that shows the character,” said Allardyce.
From a tactical point of view, I’ve simplified the game and I’m man-managing the players. I brought some experienced and talented staff with me, along with the rest of the staff – I want everyone to get better. To turn around so quickly is a great credit to everyone. Not just me.
“What looked like a desperate situation is becoming a comfortable one. And now people are looking up the table rather than down. The last clubs I took over were tougher to turn around initially,” he said.
Everton will play Chelsea next on Saturday and this will define if Everton can really battle it out with the third-placed champions.
“We’ve got to try and get a result against Chelsea and then we’ll really know what we’re made of,” said Allardyce.
“With this confidence, if we can get a result, we’re on the right track for finishing as high as we can.
“Let’s see if the lads can pull one out of the bag.”