The Toffees manager believes that the Ireland international has been an integral part of the club's positive displays since his arrival from Old Trafford
Speaking ahead of the Toffees' fixture against Chelsea, which is due to take place on Sunday December 30 at Goodison Park, Moyes said that Gibson had performed 'really well' but also stated that there was more to come from the Irish international.
"He's been really important since he came [to the club]. If you see his record since he came, it's been very good," Moyes told reporters.
"He's done really well, we've been pleased with him. I always say I want a bit more from him. I think he's got a little bit more in the locker to come and hopefully we will get that out of him in time," added the Everton boss.
Gibson is currently struggling with a thigh strain which makes him an injury concern for Everton's tie against Chelsea, and he will undergo an assessment in order to evaluate his fitness prior to the game on Sunday.