Six months is not enough, Moyes claims
Two goals from Manuel Lanzini and one from Marko Arnautovic helped to secure the Hammers three points that left them 13th in the final standings, and Moyes said his side's performance showed a marked improvement on their play from when he took over in November 2017.
Although they ended the season with a run of three unbeaten games, prior to that sequence West Ham had achieved just one win in eight, with Moyes offering no clarity on whether he would be allowed to remain in his job next season.
"I can't say because I am no further forward," he told reporters. "We will meet and talk next week. We haven't arranged a date or a time.
"I don't think anyone can make a huge difference in six months but every manager now is in transit, there is no long period of time for anyone.
"I speak regularly with the board. I'd like to change things but maybe not everyone is happy with change."
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West Ham were in the relegation zone when Moyes took over at London Stadium and he said that he hoped he would be given the chance to build on the progress made in his first six months at the club.
"When I came in I said they didn't run around and you look at today, you think 'wow'," he said. "When you have good performances you might see there is something to build on.
"I thought we played really well with the ball. I go back to when I took over and was trying to get results by hook or by crook, there was a lot more structure today."
When asked for his thoughts on taking three points against his former club for the first time in six attempts as a manager, he said: "I can only ever talk good of Everton and what they are like.
"They gave me great backing."
Moyes' opposite number Sam Allardyce spent just three and a half minutes summarising his thoughts on the game to reporters, and rebuffed questions about Wayne Rooney's future amid rumours the former England striker is set to join DC United in MLS.
Allardyce said: "We could have easily acquired a draw but I think the quality of West Ham's two second-half goals was the difference."