Two 1-0 wins in warm-up games and a 1-1 draw in their opening Group D clash in Donetsk confirmed that Hodgson, installed as manager on May 1, is building his team from the back - and building confidence at the same time.
Unbeaten in three outings under their new manager, England avoided further injuries and showed enough flashes of promise to justify thinking about making the last eight.
But to do that, warned goalscoring defender Joleon Lescott, they will have to continue working hard and creating the momentum that can carry them through difficult games against Sweden and Ukraine to reach the quarter-finals.
"Overall, it was a good performance - and a very accomplished one as well," said Lescott, whose headed opening goal after half an hour was followed by Samir Nasri's equaliser nine minutes later.
"Obviously, France are unbeaten in a lot of games - now 22 - so we knew it was going to be tough. Before the tournament, France were one of the favourites to win the competition never mind our group.
"So I think we have to be positive and take confidence from a good result," added Lescott whose goal was the first he has scored for his country.
"On paper, it looks as if we have come through our hardest game with a draw but we know Sweden won't be easy and Ukraine are the home nation. They are going to be difficult games.
"The key is to keep it in our own hands and go into each game with everything to play for, to build momentum and keep improving.
"Not losing is the thing that breeds confidence and then it grows and grows. It was important for us to do that and we achieved our target.
"Maybe we fell away a bit after we took the lead, but it was still a good performance and we know what to do for Sweden now."
Reduced expectations have enabled England to work on their shape and style without the usual hype but, after Sweden's 2-1 defeat by Ukraine, hopes will rise with the knowledge that a win in Kiev could put them on the brink of going through.
"It was a building performance for us," said Hodgson. "We are improving all the time. We are going to be fitter and a better team with each game that we play, so I hope that brings us the results we need."
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