The Goodison Park club made a blistering start to the Premier League season which led to many pundits tipping them to finish in the top four and qualify for the UEFA Champions League.
But Saturday's share of the points with Norwich City was a fifth draw in a run of one win in seven games which has seen fifth-placed Everton fall four points behind the top four.
And Moyes has implied there may be elements of complacency slipping into his squad amid some injury problems and he could look to address this issue, particularly in goal, in the January transfer window.
"We need competition in all areas and that extends to Tim Howard in goal, absolutely," he said in The Guardian.
"Everybody has got to make sure they are at the top of their game. We have got to make sure we are not continually giving away things, making mistakes that are leading to goals which cost us.
"We need competition and there hasn't been enough of that in the last few weeks when we've had a few injuries. It is something we have to think about in January."
Moyes also wants more from striker Nikica Jelavic, who has scored just once in seven games for club and country and was involved in a row with Leighton Baines against Norwich as frustrations boiled over.
"I think realistically we have got a team who can possibly challenge for the top four still but we would need most of our players to be fit to be able to do that," said Moyes in the Daily Telegraph.
"Baines did go over to speak to Jelavic and, yeah, I think we are looking for a bit more from him at the moment, because we are a bit short.
"He is looking to try to find a bit of form and we could look to get him a bit better service."