England manager Southgate and his assistant Steve Holland on Monday signed new contracts until December 2024.
The former Three Lions defender will lead his country in the World Cup next year and attempt to go one better by winning the next European Championship in three years' time.
Southgate led England to the semi-finals of the 2018 World Cup in Russia and they were beaten by Italy in the final of the rescheduled Euro 2020 at Wembley in June.
The 51-year-old feels he has unfinished business after coming so close to winning a first major tournament since 1966.
He said: "The reality is we had already agreed [to stay]. Although we didn't feel it right to announce things before we qualified for the World Cup we knew what we wanted to do.
"In football, when you have a team who are a good team and you've done a lot of the work culturally, where they are at a point where they can challenge, you want to have a go.
"We still believe that's possible [to win a trophy]. It's an aim we should have as a team. To step away at a moment when the next few years could be very exciting, I think that could have been very difficult to live with.
"If this contract is the last I think I would still only be 53 by the end of it and I'd like to think I have a long life to live after that."
He added: "There was never a consideration that I wouldn't go to Qatar. What I wanted to be sure of was that I was wholly committed beyond that.
"It is something I feel deserved time. The way it ended for us [in Euro 2020] took a lot of emotion and energy. We were so quickly into World Cup qualifying, I wanted to allow that time for consideration."
Southgate feels England have learned from the heartbreak of falling short in the past two major tournaments.
"I think the players are at a point now where they're getting lots of big match experience, we've had a couple of near misses which, while painful, are games we've learned so much from," he said.
"We have to be at the latter stages consistently. It's the process of winning for any team, we're starting to get to those latter stages consistently and we must continue to do that.
"The response from the fans and the nation has been incredibly fulfilling. There was a period I heard people saying they didn't care about international football but I think that was a case of being hurt.
"It's been unbelievably rewarding [uniting the fans]. The players care about playing for England, there are moments when they break, when they're under huge pressure, but we've been able to refresh the team and bring new players in and whoever we have selected has performed at a really consistent level."