The Hammers moved above Liverpool into fourth with a win over London rivals Crystal Palace, with Tomas Soucek scoring twice before Craig Dawson headed home his first Premier League goal for the club.
Moyes has turned West Ham from relegation battlers into European contenders and, while urging calm over what can be achieved this season, says there is still more to come from the east Londoners.
"We are not scratching the surface yet," said Moyes. "There is so much more we can give and do. Part of a manager's job is to raise expectations but I won’t be stupid as West Ham have done it too often and maybe got it wrong.
"We'll be steady and sensible but ambitious, I want to compete at top of the league – the players are giving me every hope and I can sense they want more.
"I want to grow a club, or help… which will do it steadily. It’s rare you can get here from where we’ve been and stayed there. You have to do it in stages but we’re aware it could easily go."