The England defender, 30, currently wears the armband in Phil Neville's absence and looks set to take over the role full-time when the club skipper leaves Goodison in the summer.
But manager Moyes said: "Jagielka is always bouncing up and down in the corridor and you can hear him when he comes on the training pitch.
"It is good to have that type of character.
"But there are other bits I want from him. He needs to be responsible, he needs to be able to communicate and I want him to help me around the club.
"He is beginning to understand that more. Phil Neville is the best at putting the team before himself and, in a way, Jags is going to have to do that as well.
"He has got to help me in what we are doing and where I need the club to go. If you have a good skipper it is a big help to the manager.
"I don't want him to be Phil's clone, but I want him to look at some of his traits and I think if he does that it will benefit him."