Ricky Sbragia stood down as manager of Sunderland on Sunday after the Wearside club safeguarded their place in the Premier League.
The Black Cats were beaten by Chelsea at the Stadium of Light, but they survived as north-east rivals Newcastle United and Middlesbrough also lost and went down.
Sbragia told Sky Sports: "I've had a chat with Niall (chairman, Quinn) this week and I think we need a bigger name to carry the club further. I've done everything I needed to.
"I was given the opportunity to manage the club - and I've kept them in the Premier League. I would like to have done it better but I'm really pleased we stayed up."
Sbragia, who took over following the resignation of Roy Keane in November, continued: "I've loved it.
"There's been a lot of pressure - we started so well and maybe got carried away, losing our focus - but I always thought we were good enough to stay in the Premier League.
"I would like to have finished it today with a win but I'm pleased for the players and my staff. We've got a foundation and I'll be thoroughly enjoying seeing Sunderland move up that table.
"I'll come back in the next couple of days and tidy things up but I'll enjoy tonight and I'll look forward to some time on my own reflecting on what's happened in the last five months."
Quinn was full of praise for Sbragia on Sky Sports, adding: "Ricky stood up to be counted when I needed him.
"He was given the job of keeping us in the Premier League - he made it hard after such a good start but he got us there.
"He's staying at the club - he's got a job for life here. The majority of his coaching years have been spent here and I might just have one of the best people around to help me with player recruitment.
"I'm sure all the fans will raise a glass to him tonight - he's been superb for us."
Quinn continued: "Today is an emotional day - we've got over that line.
"Ricky has said what may happen needs a bigger name, someone to change the mentality.
"I've not thought of who yet but Ricky and I have only come here (to the situation) in the last 24 hours. I will when It all settles down - but we'll take our time."
As for talk of a proposed takeover by businessman Ellis Short, the Irishman added: "It's not quite there yet - but it was all about football this week.
"I didn't want to throw confusion into the arena - I can't begin to explain what the last five or six months have been like.
"Now is about taking a big deep breath - hopefully we'll have something positive to say about the club shortly."
Quinn also sympathised with the plight of the club's north-east rivals, adding: "I'm not gloating because we have stayed up, it's relief more than anything.
"I know what Steve Gibson (Middlesbrough chairman) and Alan Shearer (interim Newcastle manager) are going through - we're very relieved to get over the line."