The Toffees’ first match after Moyes replaced Sean Dyche saw them beaten 1-0 by Aston Villa, a third successive Premier League loss that left them with a record of one victory in 12 outings.
They have gone unbeaten in seven league matches since, winning four, and are 15 points clear of the relegation zone in 16th ahead of Saturday’s trip to 17th-placed Wolves.
Moyes said at his pre-match press conference his players had done a “remarkable job”, and added: “I think they have (surprised me).
“I think they probably surprised all the people in this room as well, how well they have done over recent weeks, and probably surprised most of the Evertonians who have come to the games.
“I think now we are seeing players who are certainly stepping up to better levels. We’ve showed a little bit of consistency. I think the bit now is can you go further, what you’re going to do next, how can we keep it going, and I’m just talking about little bits of progress every week.
“We’re certainly not turning water into wine, but if we can keep moving slightly along the road and feeling we’re a bit better, I think that’s the right way to go.”
Moyes – who stressed, with Everton on 32 points, he was “certainly not thinking we’re safe” – also emphasised that while wingers Jesper Lindstrom and Jack Harrison had done “a really good job in recent weeks”, he needs assists and goals from them.
Both players are yet to register on either front this season, and Moyes said: “Let’s be fair – you won’t have longevity if you’re not going to get me assists and goals. You have to find ways of coming up with those numbers.
“I think both have done well but I need goals from them, I need assists from them. They both need to improve greatly on that.”
Another individual Moyes spoke about looking for more from was his fellow Scot Nathan Patterson, who returned to the matchday squad after injury when Everton drew 1-1 at Brentford nine days ago.
“He’s been here (three) years now,” Moyes said. “I know he needs to step up. I’m hoping the next month or two we get a chance to work a bit closer with him.”
Everton have a considerable list of players still out because of injury – Dominic Calvert-Lewin, Armando Broja, Iliman Ndiaye, Dwight McNeil, Orel Mangala and Seamus Coleman – but Abdoulaye Doucoure is available after missing the Brentford game due to the birth of his son.
The game, which Everton go into after a warm-weather training camp in the United Arab Emirates, is an 8pm kick-off, and Moyes said with a smile: “Normally a bottle of red wine down by that time on a Saturday night maybe! But I do think we’re all becoming a little bit more acclimatised to different kick-off times.
“I think if you’re a traditionalist then we probably don’t see 8pm on a Saturday as a good footballing time, but that’s the way it’s gone…we have to go with it. Obviously it will ruin some people’s Saturday night out!”