MK Dons brought in 10 new players last summer, including former Sheffield Wednesday forward Callum Paterson, while Warne joined as manager.
Warne has won promotion to the Championship four times, with Rotherham and Derby, and his appointment hinted at the club’s long-term goals beyond this campaign.
“I think we’re going to be a bit more ambitious than survival (next season),” he said.
“We want to go up and be competitive and give ourselves the best chance of being in the top six region, that’s where we want to be.”
While declining to set a target for next season, Hart admitted the club’s ambition lay in the second tier and that Dons owner, Fahad Al Ghanim, was “invested in taking the club forward”.
“We should be definitely (in the Championship) and that’s where we want to be,” Hart said.
“There’s a lot of work to get there, but we want to realise the ambition, not just the stadium, but the size of the city and the region, and we want to give people a club to be proud of and come and support to get involved in.
“We want 20,000 to 25,000 people in that stadium week in, week out, but we’ve got to deliver the football and we’ve got to get it right, and that’s what we’re trying to do.”