Dyche says Everton exit came at 'the right time'
Dyche was sacked just hours before the FA Cup tie with Peterborough last Thursday by Everton's new US-based owners, The Friedkin Group, with the Liverpool club just one point above the Premier League relegation zone.
Within two days, David Moyes returned as Everton manager, with the Scot having previously held the job at Goodison Park from 2002 to 2013.
Dyche, in a statement issued by the League Managers' Association, said: "It was an honour to manage Everton, a football club with a significant heritage and an enormous following in Liverpool and all over the world, through one of its toughest periods in its history.
"Despite the challenges we faced, I wanted to ensure that the club's narrative was focused on the positive direction it will take in the future and that the team could focus on competing in the Premier League in the present."
The 53-year-old Englishman, who thanked his staff, players and Everton fans for their support, added: "The right time has come to leave the club, with the team in good shape.
"David is a manager who I have great respect for and I believe he and his staff will help to push the club forward again and build on the stable footing we have implemented in the past two years."
Dyche took charge at Everton in January 2023, after nearly 10 years as Burnley boss.
Everton, ever present in English football's top flight since 1954, survived relegation at the end of the 2022/23 season by just two points.
They also overcame an eight-point penalty for breaches of the league's profitability and sustainability rules to finish 15th last season.
Dyche added: "I wish everyone well going forwards including the new owners, who I hope will take the club to the next level and deliver the success which every Evertonian wants.
"My part as a custodian has been played and I will forever take great pride in that."
Under-18s coach Leighton Baines and club captain Seamus Coleman took charge of the Peterborough tie, which Everton won 2-0.
Moyes's first game of his second spell in charge is a Premier League fixture at home to Aston Villa later on Wednesday.