The 43-year-old looked ready to become the successor to Neil Lennon at the Easter Road club, only for discussions to hit a snag at the 11th hour.
Paul Heckingbottom, who was in the running alongside the ex-Oxford United boss, was hired as the new manager just a couple of days later.
It had been reported that the reason behind the breakdown was Appleton’s demands over a severance package.
Speaking to Sky Sports, the former Portsmouth midfielder would neither confirm or deny those stories, saying that he wasn’t prepared to reveal the reason he didn’t become Hibs boss.
He said: “I think if I answer that honestly and truthfully, I would probably come across as being arrogant and disrespectful, which is a million miles away from the truth.
“I think from my point of view, I will continue to look for the right thing for me, whether that is to go in somewhere as an assistant manager or a manager.
“(I will) look for that club that shares the same ambitions that I have got myself.”
Asked if he was disappointed that the move had transpired, he added: “Opportunities at clubs come and go all the time.
“You have that initial disappointment but you can only take a situation so far.
“I got some fantastic advice, not too long ago, about making sure everything is comfortable with yourself.
“If it isn’t quite right, for whatever reason, it is probably the wrong one for you.”