F1 CEO Stefano Domenicali also stopped by the junior Red Bull team's Albert Park base to offer Hadjar a supportive gesture.
"It was very kind of them," said Hadjar, the 2024 Formula 2 runner-up. "They've seen many drivers go through such lows, and to come to me like that really is very kind."
Hadjar called his failure to reach the slick Melbourne grid "simply embarrassing". "That was really the worst moment of my life," he added.
Red Bull's famously stern adviser and driver overseer, Dr Helmut Marko, felt Hadjar's shame stemmed from the wrong reason.
The 81-year-old remarked that the latest Red Bull rookie in Formula 1 "put on a tearful show" in Australia, chuckling: "That was a bit embarrassing."
Ex-F1 driver Ralf Schumacher took issue with Marko's stance.
"I wouldn't have cried myself," the German told Sky Deutschland, "but this is a very hard moment for a young driver.
"It was his first race, and up until that moment he had put in a fantastic performance. Then it's awful when you let your team down at the end.
"At Red Bull, you quickly become afraid of your future, given Helmut's reputation. And honestly, I don't know what's embarrassing about crying."
Racing Bulls CEO Peter Bayer, however, backed Hadjar. "I think this can happen to anyone in these conditions," he said. "One small, minimal mistake, and it's all over.
"Now we're looking ahead. We have another race next week."
Hadjar responded: "It's nice to see that I'm getting this support from the team. It means a lot to me."
Another former F1 driver, Heikki Kovalainen, sided with Marko.
"I admit that I don't really like these kinds of crying stories," the Finn told Viaplay. "This is a tough game, and the mental edge has to be hard. And crying won't help."
Hadjar reflected: "I've had similar lows, but this one is particularly tough.
"My path to Formula 1 wasn't easy - it was pretty brutal. I'd say I have a strong chin, but this one really hits me."
