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Hodgson's side lost their opening games 2-1 to Italy and Uruguay, and Costa Rica's 1-0 win over Italy on Friday meant the thin mathematical chance England had was gone.England's first exit at a World Cup group stage since 1958 has put the manager's position in the spotlight, but Hodgson has insisted he will not resign and the Football Association has backed him.
Its support was echoed by both Baines and Hart, who insisted that Hodgson remained the right man for the job.
"We are going in the right direction," said Everton's Baines. "There are some improvements to be made still, but I think we have made some positive strides.
"The manager has given everyone a go, and you would imagine some of the young lads are going to be around for many, many years. Ross [Barkley] had some magic moments against Italy and Uruguay.
"Continuing that progression will be positive. The easiest thing to do sometimes is to be critical and, with the position we are in, I suppose you have to accept that up to a point. But there were some positives in both the games, even if we will only really see them maybe further down the line."
Manchester City keeper Hart said everyone in the squad was "proud to play for Roy Hodgson" and described him as "a great manager."
"He's a very passionate man and someone that I have got an awful lot of respect for, and I hope to continue playing for him," he said.
And he insisted that, despite the huge disappointment of England's short-lived Brazil campaign, he also believed the team had progressed since Hodgson took over before Euro 2012.
"I imagine people will say we're not good enough, but we don't feel that inside the camp," he added. "We've got bags of quality, bags of people coming through, great leaders and great experienced players in that dressing room.
"But we've just come up short in two games, and you can't come up short in two games at the World Cup."
Should Roy Hodgson be sacked?
Stevie Nicol discusses whether he thinks Roy Hodgson should be sacked for not being able to advance England out of the group stage.
Le Saux lends Hodgson support
Former England defender Graeme Le Saux has backed manager Roy Hodgson to remain as national team manager.