Hallgrimsson has taken sole charge having shared coaching duties with Lars Lagerback, who retired after the tournament, at Euro 2016.
"The players are now legends in the eyes of Iceland's people," Hallgrimsson told The Guardian.
"They will be remembered in the same way England venerates its World Cup winners from 1966; they have given the currency of our football a huge boost and it will extend beyond sport too, as a wonderful advertisement for a country that is now on the map like never before.
"Soon I will sit down with my staff and other members of the Icelandic FA to look, with the benefit of distance, at how we achieved what we did at Euro 2016 – what was good and what was not so good."
Hallgrimsson suggested Iceland supporters will now have increased expectations for the national team after seeing their success at Euro 2016.
"Icelanders are demanding – we will now be expected to qualify for every major tournament," the coach said.
"There will not be too many changes on the pitch as we approach the World Cup qualifiers; we are lucky that most of these players are of a good age and few will step out of the squad in the next two or three years.
"Even though some of our younger players did not play much in France it was fascinating to see how they grew within the squad in a tournament environment, and that bodes well."
On the reception his squad received upon returning to Iceland, Hallgrimsson added: "Sometimes pictures say more than words ever can.
"The scenes that greeted us when we arrived in Reykjavik on Monday, only a few hours after our quarter-final against France had finished, will stay with me forever.
"As we drove the 40 minutes from the airport to the city centre, huge numbers of people lined the streets and then, finishing up on Arnarholl, thousands more awaited us.
"You could only feel intensely, overwhelmingly proud; the speeches we heard, from our president, prime minister and others, would have softened any heart and brought home the scale of what our boys did this summer. None of us could ever, for a single moment, have envisaged the reception we were given."