Whoever does take over will assume control of a team who are tireless in terms of their work-rate but one which needs extra quality and ruthlessness in front of goal.
Palace survived two early Everton chances when Leon Osman sidefooted wide and Kevin Mirallas had a driven shot smartly palmed around the post by keeper Julian Speroni.
However, the Eagles should have gone into the break in front but for some poor finishing.
John MotsonBBC Match of the Day commentator
"The threat of Romelu Lukaku was pretty well contained by Danny Gabbidon and Damien Delaney. It was a thoroughly enjoyable game. It felt like a victory for Palace. The Palace players put in a huge shift. It's at least a little bit of light at the end of the tunnel."
First Dikgacoi evaded his marker to meet a Thomas cross but the midfielder's header ended up being more of a clearance than an effort on goal.
Thomas was back in the action again when his long-range shot was tipped over by goalkeeper Tim Howard at full stretch, moments before Chamakh sent a powerful header wide from Barry Bannan's cross.
Palace's regret at those missed chances increased when they wasted a glorious second-half chance.
Thomas was played clean through on goal - beating the Everton offside trap - but the ex-West Brom winger's effort saw him chip Howard and find the side-netting from the edge of the box.
The home side almost paid the price for their profligacy when Eagles striker Cameron Jerome inadvertently flicked on a corner which Jagielka headed against the crossbar.
Palace had to endure some tense moments before the final whistle but managed to hold on and end a run of seven straight Premier League defeats.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.