Celtic were very much second best to Rangers last season and, as has been said hundreds of times in Australian media in recent weeks, such a scenario is unacceptable in that football-mad city.
Time is of the essence. The former Socceroos boss has plenty of doubters in Scotland and needs to hit the ground running to get fans and media onside.
How he performs in the transfer market will be an early sign of what he intends to do.
Postecoglou is sure to have a more global mindset than his predecessors and be familiar with the best talent there is in Japan, Australia and the rest of Asia but for his first major deal he could do a lot worse than turn to a familiar face - that of Aaron Mooy.
The reaction to a sudden influx of players from Australia and Asia may lead to more groans among those Celtic fans who are sceptical of the standard of football in the world’s biggest continent.
That is a debate for another day but, anyway, Mooy would be different.
After all, the 30-year-old has performed, and performed well, in the English Premier League.
After his time with Huddersfield Town, he went to Brighton & Hove Albion. Needed just a little time to settle on the south coast, the midfielder established himself as a vital player as the team stayed up.
It was only a big money offer that resulted in last August’s move to Shanghai Port in China.
There has already been talk in Shanghai of a possible Celtic approach and the club would be unlikely to play hardball.
There is the question of his salary, which is significant, but the transfer fee would not be.
Chinese teams understand that the current situation in the country is tough for foreign players.
All players are confined to two ‘bubbles’ in the cities of Guangzhou and Suzhou, and leave hotels only for training and games.
That is one reason Graham Arnold did not select Mooy for the ongoing 2022 FIFA World Cup qualifiers in Kuwait.
"Aaron being in China, being on his own, away from his family, he hasn't seen his wife or kids for four months," Arnold told Fox Sports in May.
"The family haven't been able to get into China. Aaron, he felt he had to see his family. I support that 100 percent.
"Family is number one in life and Aaron is so committed to the Socceroos and was put in a position where if he didn't go and see his family now he wouldn't have seen them for 12 months ... I can't wait to see Aaron back in camp in the third phase in September."
Arnold would surely be happy if Mooy was a Celtic player under Postecoglou by the time the third round starts.
Mooy’s wife is Scottish and being able to live with his family while playing football would be a major positive and lead to much better work-life balance.
In football terms, Mooy would arrive in Scotland as one of the best players in the league, the box-to-box midfielder who knows Postecoglou well and can be his leader on the pitch.
In China, he has come to take on more attacking responsibilities than in the past but would surely settle in any role in Celtic’s midfield.
He has the mentality too. Celtic are a big club but Mooy has impressed wherever he has gone.
His move to Shanghai to become one of the major stars at an ambitious club in a growing league is not to be sniffed at either.
There are pressures on such players as well as the challenges of adapting to a very different life on and off the pitch.
Postecoglou knows that while he may want to build a long-term project at Celtic, he won’t be able to if he doesn’t produce in the short-term.
Mooy is a risk-free signing that helps with both. He will hit the ground running and could be a star for years to come.