Gareth Southgate's side made the last four for the first time in 28 years, but a second defeat in the tournament to Belgium means they ultimately lost three of their seven games in Russia.
"We closed it for sure (the gap to best top teams). But today shows, and some other games show, that there is still room for improvement," said Kane.
"We?ve said that all along ? we're not the finished article yet, we?re still learning, we're still getting better.
"We don't want to wait another 20-odd years to get to the semi-final in big matches. This is the level we've got to stay at now. We've got to improve, we've got to get better, but it will come."
Kane failed to find the net in England's final three games of the tournament, but still remains on course to win the Golden Boot thanks to his six goals against Tunisia, Panama and Colombia.
"Obviously I'm disappointed I couldn't get a goal in the last few games," added the Tottenham Hotspur striker.
"Sometimes it goes for you, sometimes it doesn't. But if I win it, it'll be something I'll be very proud of."
Best-ever World Cup finish
Belgium's victory over England in the World Cup third-place playoff ensured their best-ever finish.
Goals from Thomas Meunier and Eden Hazard secured a deserved 2-0 victory in Saint Petersburg and meant Belgium bettered their performance at the 1986 World Cup in Mexico, where they finished fourth.
Belgium, who suffered a painful 1-0 defeat to France in the semi-finals, also beat Gareth Southgate's England in the first round in Russia on their way to topping the group before beating Japan and Brazil.
"It's all about the achievement. This is Belgium's best finish at a World Cup. The players deserve that and they wanted to make the country proud," said coach Roberto Martinez said.
Southgate's men were forced to settle for England's joint-best performance at a World Cup abroad, matching that of the 1990 generation in Italy.
Tottenham's Kane remains the overwhelming favourite to win the Golden Boot with six goals, although he admitted his frustration at failing to find the net in England's final three games.
He said despite England's run to the semi-finals, where they were beaten by Croatia, the young team are still not the finished article.
"We closed it for sure (the gap to the top teams)," he said. "But today shows, and some other games show, that there is still room for improvement."
"Obviously I'm disappointed I couldn't get a goal in the last few games," added the Tottenham Hotspur striker, who is aiming to become just the second England player after Gary Lineker in 1986 to win the Golden Boot.
"Sometimes it goes for you, sometimes it doesn't. But if I win it, it'll be something I'll be very proud of."