The 59-year-old Spurs boss led the North London-based squad to a 1-0 victory against Manchester United at San Mames Stadium in Bilbao, Spain, on May 21 - ending the club's long 17-year wait for a major trophy, as well as receiving Champions League qualification.
However, he still has the shadow of team finishing 17th - their worst result in Premier League history - following a 4-1 defeat to Brighton hanging over him.
But Postecoglou is determined to keep pushing for Tottenham to produce more success stories.
The manager said at his post-match press conference: "I'll be honest, I've been finding it really weird talking about my future when we've done something unprecedented.
"I've had to answer the questions because no one else at the club is kind of in the position to do so, I guess. But I can't answer that question about me and my future, and part of me is kind of thinking, well, why have I been asked that question? But it is what it is.
"And I've got no doubt, though, that this could be a real sort of defining moment for this club because wherever I've been, I've kind of made an impact where I've brought success to a club that hasn't had for a while, and you just have to look at those clubs' trajectory even after I've left, they're still competing for things.
"I really think this is a moment in time where this club can really push on and be a real contender for honours on a yearly basis.
"My gut feeling is that I feel, right now, that I've done something that no one believed I could. And I shouldn't be here, sitting there, talking about it. That's my feeling, but it is what it is.
"And probably I'm talking now because I'm tired, mate. It's done. I don't have to think about another game. I just want to go on a break and enjoy my family and reflect on being part of something unbelievably special."
He continued: "I've been fortunate to be in this position a number of times, but seeing those scenes yesterday and the people who've reached out to me and talked to me about how it's affected and what's happened the last couple of days, that's just a beautiful thing and I just want to enjoy that.
"To be honest, the rest of it, I don't even want to be talking about it. I just think it's unnecessary."
What Postecoglou is "so confident" about is the possibilities of more success stories, and the determination for him to "push on".
However, the Spurs boss has said he is waiting to see whether he will be in charge of continuing the drive for more glory in the future "happens or not".
He added: "You're right to ask the question, but you're asking the person who can't give you that answer. And I guess, even for you guys, you wouldn't be asking it if there wasn't a doubt. But I can't answer it. There's nothing I can say that can answer that question. Other people can, so from my perspective, it doesn't diminish the achievement.
"I'm so confident about what we can build at this football club, and I want to push on and take it to the next level, and we'll see whether that happens or not."
