England lost five of their six games before the tournament but came within three minutes of reaching the final.
“We’ve shown stuff that fans can hopefully get behind and be proud of,” said Care.
“The support we’ve had here and at home has been amazing. This team will go on to bigger and better things, I’m sure.”
A late Handre Pollard penalty condemned England to a 16-15 defeat, as the World Cup holders advanced to a final against New Zealand.
England emerged with credit, and Care said: “There was a lot of stuff said about us before and hopefully we’ve changed some perceptions, maybe got people believing in us again.”
Steve Borthwick’s side had the most players aged 25 or under among the four semi-finalists, with his squad now facing Argentina in the third-place play-off on Friday.
That will be a repeat of their opening fixture, which England won thanks to 27 points from the boot of fly-half George Ford.
“It will benefit every one of these players and this team as it moves forward,” Borthwick said of the game.
“It says there are a lot of good young players around that are benefiting from this experience. I think there are some experienced players who have plenty of years in an England shirt ahead of them.
“I want players playing big games at World Cups, I want them having that experience.”
Full-back Freddie Steward and second row George Martin, who are both still only 22, impressed against the world champions, with another Leicester Tiger in Ollie Chessum, 23, coming on to replace Martin in the second half.
Flanker Courtney Lawes praised the maturity of Martin, drawing comparisons with his own international career after making his World Cup debut in 2011.
“He is 22, the same age as me when I had my first World Cup,” said 34-year-old Lawes. “I think he is going to be here for a long time and he has some incredible work he is going to get through.”
Ahead of England’s final game in France, Care said: “We’d love to have been in the big dance but we’re not. The next best thing you can do is to finish third and try to make more people back home proud of this England team.”
Among the young talent, some 17 members of Borthwick’s 33-man squad had previous World Cup experience and 16 were in the squad when England reached the 2019 final under Eddie Jones.
Such experience could now mean a changing of the guard after the competition.
“I think it’s really important for us to finish on a high,” Lawes added.
“We’ve got a lot of boys that may or may not play, obviously we don’t know the team yet but a lot of boys that are on this tour won’t represent their country again so it’s very important we finish where we started.”
Former England fly-half Paul Grayson believes the experience of coming so close in a World Cup semi-final will help inspire the next generation.
“This isn’t the end, it is the very early days of that new journey,” Grayson, a 2003 World Cup winner, told the Rugby Union Daily podcast.
“That is the stuff you build on for the rest of your career. As that is the baseline to build your international career and rugby on.
“That has set the bar, so the challenge is now to come back, hit it again and go above it. That is how you develop a champion team.”
