Roy Hodgson says England's 2-1 loss to Iceland and exit from Euro 2016 will cause longer-term "damage" to the team.
And Football Association chief executive Martin Glenn said it is now "imperative" to find out why England are "brittle" at tournaments.
Hodgson, who resigned after Monday's game, repeatedly questioned why he had to attend a news conference on Tuesday.
But he said: "One particularly bad game has caused a lot of damage to me personally and the team going forward."
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Hodgson, 68, added: "We have a major bridge to repair - had we played better last night that might not need repairing."
England won just one of their four games at the tournament in France, qualifying second from their group behind Wales to reach the last 16, where they suffered the humiliating defeat by a nation with a population of just 330,000.
Hodgson said there were no "magic answers" to explain England's performance at the European Championship, where they also drew with Russia and Slovakia and beat Wales with a last-minute winner.
Glenn did not rule out appointing a foreign manager as Hodgson's successor, adding: "We are looking for the best person, not necessarily the best Englishman."
But he said the key thing to address was England's regular failure, which has seen them reach only three semi-finals at a major tournament since they won the 1966 World Cup as hosts.
"We need to punch our weight in tournaments in a way that we have not done in 50 years," added Glenn.
"When we get to the business end of a tournament, England seem brittle and we need to understand why that is."
'I do not know what I am doing here'
Hodgson read a statement to announce his resignation after the Iceland game which he felt was "sufficient" - and four times during Tuesday's news conference he said he should not need to face the media again.
"I am still a little bit unsure what I'm doing here," said the former Fulham, West Brom and Liverpool boss. " I suppose someone has to stand and take the slings and arrows."
When BBC sports editor Dan Roan suggested Hodgson had been forced to attend, Hodgson replied: "I wasn't forced to come. I was anxious to make certain no-one in this room can say I was worried to face the media."
'Players did not lose faith'
Hodgson faced heavy criticism following the goalless draw against Slovakia for making six changes to the side that had beaten Wales, despite knowing a victory would mean England topped the group.
He was also criticised for the tactic of having Tottenham striker Harry Kane take corners during the tournament and starting Manchester City forward Raheem Sterling against Iceland.
During Tuesday's news conference, England captain Wayne Rooney released a statement saying reports the players had lost faith in Hodgson are "completely untrue".
When asked about the reports, Hodgson added: "If it was true, they disguised it very well from the players and coaching staff."
Glenn described the dressing room as a scene of "devastation and personal grief" after the defeat at the Allianz Riviera in Nice.
He added: "Let's scotch that one - it's not about a lack of passion amongst our players."
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England scored only four goals in four matches at Euro 2016 and Hodgson felt his side were not "ruthless" enough in France.
He said: "We did not play well last night and I take full responsibility for that.
"We showed signs of good football in the first three games, which gave us confidence, but last night we didn't reproduce.
"I'm disappointed. I didn't see it coming. I had no indication that we were going to play that badly."
He added: "I'm sure those players will live up to expectations and one day I'm sure we'll see England do well in a tournament - I'm hoping in 2018 [at the World Cup in Russia].
"They have shown they are worthy of wearing an England shirt and they have shown myself and my coaches that in this tournament."
Glenn will be fronting a three-man panel to decide the next England manager, with FA technical director Dan Ashworth and board member and ex-Manchester United chief executive David Gill.
They will be canvassing opinion from current and former managers, players and clubs.
Glenn told BBC Radio 5 live that a new manager could be in place for England's World Cup qualifier against Slovakia on 4 September - but did not rule out an interim manager being appointed.
And on England's ability to recover from the disappointment of Euro 2016, he added: "It's a dark day but I'm an innate optimist.
"If you go back 10 years and [Arsenal manager] Arsene Wenger saying he wasn't seeing the technical quality coming through at age 10 - that's not true any more.
"This team cared. How to prepare them to make them more resilient when it comes to those pressure-cooker games is the job we've got to do going forward."
Hodgson won 33 of his 56 games as England boss after being appointed in 2012, succeeding Italian Fabio Capello.
England reached the quarter-finals of Euro 2012 - losing to Italy on penalties - before a dismal showing at the 2014 World Cup, where they exited at the group stages without winning a game.
Glenn added: "I want to thank Roy. The set-up now is much stronger than the one he inherited. Iceland is not your legacy or epitaph, we have much to be grateful for."
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