The German boss won his first silverware since becoming Liverpool manager when he guided the Reds to a 2-0 European final win over Tottenham at the end of last season, goal.com reported.
Liverpool's European triumph came after a memorable campaign that saw the English side famously erase a 3-0 first-leg deficit to La Liga champion Barcelona with a 4-0 second-leg comeback to make the final.
And while there may be some thought the Reds' success will leave them satisfied, the manager claimed the opposite – their victory only leaves them hungry for more.
"Nobody should be worried about us being complacent," Klopp told reporters.
"There are so many reasons for staying greedy. We tried so many things to win and finally it worked out.
"The Champions League is the biggest cup that I see in football apart from the World Cup. Absolutely massive. But you now want to have it all the time.
"Nobody should be worried. Nobody is satisfied and we are not ready to go home. We have won something and got it in our trophy cupboard, but our attitude does not change.
"I couldn't have imagined it when we sang the year before at the funeral party that, 'We bring it back to Liverpool' and it would work out. We got there.
"So let's try to get there again next season. We will not even think about stopping."
The Champions League triumph sparked predictably wild celebrations in the stadium in Madrid and back home on Merseyside.
And Klopp admits he was taken aback by the scenes, including on an open-top bus tour, which fed his "greed" for more trophies at Anfield.
"It was absolutely overwhelming," he said.
"It was too much to comprehend, one of the best things I've witnessed in my life.
"Everyone tells you it will be something special if you win something with Liverpool and this was proper proof.
"I was thinking, 'Wow, that was brilliant!' then the bus came around the corner and you are like, 'What the heck is that?' I could not imagine the noise.
"We made so much eye contact with so many people – people in their late 60s or early 70s. There was one guy and he was like this: Boom! Boom! And he is shouting, 'I love you!'
"I was like, 'Stop, calm down! My God.' That is only one example of thousands.
"It meant so much to us and obviously meant so much to the people. It was absolutely brilliant. Even now, you wake up in the morning and realize it is still true."