The Toffees have narrowly avoided relegation in the last three seasons and currently sit just one point above the relegation zone. Their first match under Moyes' return will be against Aston Villa on Wednesday.
The Liverpool club's new owners, the American-based Friedkin Group, sacked Sean Dyche last week and have turned to Moyes in a desperate attempt to maintain Premier League survival ahead of a move to a 53,000 capacity stadium next season.
"They want to get us back on track and we all know we need a bit of a sticking plaster at the moment and we have to try to make that work," said Moyes at a pre-match press conference.
"A couple of weeks (ago) I didn't think there was any chance Everton could be in a relegation fight, I thought they would be strong enough to get out of it.
"I've come into the seat and I'm going to back that and say I believe we will be strong enough to stay away from it (relegation).
"But I'm not kidding myself or anyone else, we need everyone behind us, we need the players playing better, scoring more goals, if we are going to make that happen."
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Moyes was sacked by United after less than a season at Old Trafford and ill-fated spells at Real Sociedad and Sunderland followed.
He rebuilt his reputation, however, during two spells at West Ham.
After leading the east London club out of relegation trouble in the 2017/18 season, he returned in 2019 for a five-year stint that led the Hammers into Europe three times and ended a 43-year wait to win a trophy by lifting the Europa Conference League in 2023.
"I've come very close to coming back (to Everton) on maybe three or four occasions," said Moyes. "I've had talks, and for whatever reason it's fallen through.
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"It used to be that going back was no good, but I did that at West Ham, and we went on to win a European trophy. So we're back on the journey again."
(This story has not been edited by staff and is auto-generated from a syndicated feed.)