Sunderland manager Sam Allardyce is favored to take over as England coach following Hodgson's hasty resignation which came just minutes after England's demoralizing 2-1 defeat by Iceland
-- a defeat which knocked it out of Euro 2016.
Gerrard, who scored 21 goals in 114 appearances for England before retiring from international duty in 2014, says a fear of failure was to blame for England's lackluster performance in France, but that the right appointment could take it to the "next level."
"There's a huge fear amongst the group at the moment," Gerrard told Virtual Reality company Laduma in Los Angeles, where he currently stars for the MLS's LA Galaxy.
"The expectations are very, very high. It's a very pressured situation to be a England player, you know, the media coverage is huge. The expectation is huge and there's a very young set of players there who are not used to that pressure, but they will be used to it in years to come."
Gerrard added that the exposure of playing in big tournaments can only help improve young English talent like 20-year-old midfield star Dele Alli.
"This experience and this hurt, I think, will help all of those players moving forward because they won't want to experience it again," he added.
"They will want to improve and get better, and I think with the right manager and the right coaching staff around this group, they can take it to the next level."
Gerrard, who played 16 seasons at Liverpool where he won a Champions League title and two FA Cups, admitted that he had expected much more from his former teammates.
"I was disappointed," he said. "I had high expectations. Through qualifying they performed really well, they looked solid, they put in some terrific performances and going into the tournament I fancied them to at least get to the last eight and also have a push to the last four.
"I had that confidence; they started reasonably well, they came out of the blocks really strong against Russia. They just never got the results, and never performed to what they are capable of.
"I think everyone's sort of healing from the disappointment of the tournament, but what I can say is the FA and the payers and the new staff coming in will get it right.
"It will improve and it won't be too long before the fans are smiling again, that's what I believe. I believe in looking forward and trying to improve things and I've got every confidence that everyone involved trying to make it improve will make that happen."
Bookmaker William Hill has Allardyce as the strong favorite to take over as next permanent England manager, giving him 4/11 odds, with current U.S. national team manager Jurgen Klinsmann a distant second at 8/1.
Arsenal manager Arsene Wenger is a long shot at 25/1 odds, while his arch-rival Jose Mourinho -- who has just taken over at Manchester United -- is on at 50/1.