Captain Steven Gerrard admitted England had not played the best football at Euro 2012 and needed to work harder at retaining the ball following the quarter-final penalty shoot-out defeat to Italy.
The Azzurri dominated possession and hit the post twice, and although England were able to keep them at bay for 120 minutes, the failure of Ashley Young and Ashley Cole to convert their shoot-out kicks led to an another last-eight exit for England.
Gerrard said: "At times the football hasn't been fantastic, we haven't blown people away.
"The possession stats speak for themselves. Moving forward as a nation we do need to improve with the ball."
Despite the less-than-impressive performance, Gerrard was happy he and his colleagues had not come up short for effort.
"Everyone has given everything they've got and that's all you can ask for," he added.
"I think the supporters will agree with me that we have played for the shirt."
But Gerrard stressed: "I think we've performed to a better level at this competition than in previous tournaments."
Gerrard is keen to continue as captain and also said he was happy with his own individual performance in the tournament.
He added : "I think performance-wise I'm reasonably satisfied.
"I'm just in a zone of frustration and disappointment because we've come so close to ending that quarter-final hoodoo."
Gerrard was given the captaincy by Hodgson and the Liverpool midfielder revelled in the role.
"I've loved it, I've loved the responsibility and I've enjoyed every minute of it."
Hodgson has also been happy with Gerrard's captaincy.
Gerrard said: "I've had a brief chat with the manager, I'm available as long as I'm wanted at this level. The manager has told me he wants me to continue and carry on."
Hodgson drew comparison with the Germany team of 2006, who also suffered defeat to Italy - albeit in the semi-finals of the World Cup - early on in their rebuilding mission.
The England manager hopes his side can emulate a team which has progressed through to the semi-finals of Euro 2012, where they will face Italy on Thursday.
Hodgson hopes his side can make enough progress to qualify for the 2014 World Cup in Brazil.
"We have to keep working hard to bridge any gap that might exist at the moment between ourselves and teams like Spain and Germany," said Hodgson.
"We saw with the German team at the 2006 world Cup they went into the finals highly unfancied with a new coach and with a lot of new young players.
"We've seen how well they've kicked on from 2006. We have to take heart from that. I like to think there's good reason for optimism."