When the Premier League resumes on October 15, the Toffees have to travel to Stamford Bridge, where they have not won since 1994, and Fulham - where they have had only one victory in 44 years.
They face Chelsea again at Goodison Park in the Carling Cup and three days later host Manchester United before a further test then awaits at fourth-placed Newcastle, who are unbeaten this season.
Everton's 2-0 defeat to Liverpool, in which they played three-quarters of the match with 10 men after Jack Rodwell's controversial red card, was the first time since November 2009 the Toffees had lost back-to-back league matches.
"Take aside the sending off and we gave away two bad goals and we didn't play well enough on the day and that is something we need to improve on," Neville told evertonTV.
"We have to bounce back as we have Chelsea next and we have a run of games where we have to pick up points.
"We can't just say we will lose against the big teams and beat the lower teams because it doesn't work like that.
"We have to pick up points and play better, show a bit more composure at times and lift our performance levels.
"We will probably lick our wounds going into the international break but coming back I am sure the manager will have us fired up."
Goalkeeper Tim Howard backed his captain's call for renewed effort.
"The manager made us take pride in the fact we didn't lose back-to-back games, we were resilient and always came back," said the United States international.
"Unfortunately that has come to an end but we have the international break to lick our wounds, get better and put it behind us.
"But we have no time to rest because it is a tough stretch for us."
Both were also in agreement in their praise of 20-year-old Rodwell, whom they believe is starting to develop into an important first-team player and will not be affected by his red card.
"He started off well, getting stuck into clean tackles and he got punished for it," added Howard.
"Jack is a very strong character, he is a young kid but he has a bagful of confidence on and off the field.
"He knows he has our respect and appreciation and he will be okay."
Neville added: "Jack was outstanding and has been [in other matches].
"We have persevered with him; he is getting a run of games, is showing fitness and confidence - and is now facing three games out of the team."
Source: Team Talk
Source: Team Talk