The Toffees, who finished seventh last time around, kick off the new campaign at home to Manchester United later this evening.
"It was only a few seasons ago that the team was pushing for the top four, or just outside it," the South African told The Sun. "Since then the teams above us have spent a lot of money and the League is getting tougher.
"But hopefully we can push on from last season — I believe something special can happen at Everton this season.
"If we start this campaign the way we ended the last, we can push the teams expected to finish at the top all the way."
Piennar rejoined Everton last month following an unsuccessful year at Tottenham Hotspur.