The Hoops lost out to Rangers by one point for the SPL title and were also beaten in the final of the Co-operative Insurance Cup by their Old Firm rivals.
Lennon's men, however, ended the season on a high by capturing the Scottish Cup in their Hampden win over Motherwell.
He said: "I think the players grew as the season progressed but it is hard to gauge whether or not they have taken another step in the close season.
"Once we have three or four games we will have a good idea of where we are heading.
"The expectation level might be higher because of what we achieved last year, probably not many people expected it.
"We ran Rangers so close and got to two cup finals but we feel we have a talented group of players and we want to go one better.
"I am grateful for the opportunity to carry on as manager, first and foremost, and looking forward to the challenge of trying to improve the team.
"I want to try to get more consistency, play better football, score more goals, keep more cleans sheets and accumulate more points.
"It is the challenge ahead of us and I think Europe is also important to us, not just to myself but the players and the club.
"So we have a lot of incentives going in to the new season."