Cole was signed by Roy Hodgson, who was sacked in January as just one change in a dramatic season, with the club's new owners, Fenway Sports Group, having taken over in October.
"New owners, new manager, new coaches, new football director...it's a complete change," said Cole.
"But I came in from a club in Chelsea where we had gone through a few managers as well over the last few years, so I am used to it.
"I think continuity is so important in football. Now we've got Kenny, Damien (Comolli) and Steve (Clarke), who are as good as you'll find at doing their jobs - their track records are proven.
"It's about continuity now. The club needs that for Liverpool to move forward and for us to progress. We need to stick together."