The Toffees have lost the Scotsman to Manchester United after he was unveiled as their replacement for Sir Alex Ferguson on Thursday with Kenwright saying the pursuit for a Goodison Park successor has begun.
"We really will be out there finding the right man for this football club because it's an extraordinary football club that is worshipped by its fans," he told Sky Sports News.
"They are the people - more than anything - that I have been thinking of over the last 24 hours.
"Eleven years ago I made a decision and it was an instant decision when I met David. I don't think that can happen this time. We have got to be out there looking to see what kind of candidates can take the Club forward.
"We've had a really good 11 year adventure with David and now it is my job and the Club's job to get a worthy successor and build on what David has brought to the Club and to hopefully move on from there."