Frank Lampard's side lost 1-0 against Wolves, while victories elsewhere for Watford and Leeds dragged the Toffees further into trouble.
After a poor first half at Goodison Park - which saw Wolves equal Arsenal's 1999 record of 14 Premier League away games without conceding before the interval - the visitors took the lead four minutes into the second period.
Ruben Neves swung in a brilliant cross and Conor Coady's glancing header found the far corner.
It consigned Lampard's side to a seventh Premier League defeat in their last eight matches.
Watford's 2-1 win over Southampton meant Roy Hodgson's side climbed up 18th and level on points with Everton.
Cucho Hernandez scored two first half goals before Mohamed Elyounoussi reduced the deficit down to one.
At Elland Road, there was late drama as Leeds grabbed a first victory under Jesse Marsch.
A 14th minute striker from Rodrigo looked like it was going to be enough for a Leeds victory, before a 91st minute equaliser from Kenny McLean.
But Joe Gelhardt grabbed a 94th minute winner for a much-needed victory.
Leeds are 16th on 26 points (29 played), Everton sit 17th on 22 points (26 played) and Watford are 18th on 22 points (29 played).