The Greece-born Australian had to hastily assemble a squad together after taking over the Hoops hot seat in June and - after a difficult start to his tenure - the 2-0 Europa league win against Ferencvaros on Tuesday made it three wins in a row for the first time since August.
Celtic are fourth in the cinch Premiership, four points behind champions Rangers as they prepare for the visit of St Johnstone on Saturday.
Postecoglou would neither "confirm nor deny" that he is interested in forward Daizen Maeda, whom the manager bossed at Yokohama F Marinos before he came to Celtic, where he brought in new Parkhead hero Kyogo Furuhashi from the J-League – but is clear in what he is looking for in 2022.
"We are going early with being linked with players," said the former Australia manager, who revealed defender Josip Juranovic is back after recovering from a hamstring injury and midfielder James McCarthy is available following a knock.
"It could be a long three months, me neither confirming nor denying players coming in.
"We are looking to strengthen the squad in January and we will, we have started that process but the difference being, we are in no rush.
"Whatever business we do in January, it won't be as chaotic as it was in the last window.
"But a hugely important part of bringing players here is understanding their mentality and whether they can cope.
"I was confident with someone like Kyogo. I kind of knew him really well from Japanese football. I knew his background and I asked enough people about him as a personality, to know he would cope well.
"That is part of the process, particularly when you are talking about bringing in people in January when they are going to have to hit the ground running.
"Those are the things we will be looking into, not just their playing ability but also their personality.
"And a big part of it will be the ones who want to come here.
"When players show intent, people like Jota and Kyogo, there was never any discussion to try to convince them to come to Celtic, this was their preferred destination and this was a big part of it."
Reports claim that a deal between Celtic and 24-year-old Maeda is expected to be done, but Postecoglou said: "I can neither confirm or deny.
"There is no doubt the Japanese league is one I know well, I have an advantage there.
"Obviously I know Deizen well, I was the one who brought him to Yokahama F Marinos but I can definitely confirm we are not in advance talks with anyone to sign anyone at the moment.
"We will be casting the net far and wide, we are not going to just limit it to the Japanese league.
"With the way we have started the season and how some of our younger players have been playing, we have become an attractive destination for players but I am not going to rush into that."
Domestic cup double winners from last season St Johnstone are the visitors to Parkhead this weekend, with Postecoglou expecting a "good test".
He added: "It will be a tough game.
"Obviously, they had a fantastic year last year and I kind of followed them through their early European efforts where I thought they were impressive.
"They are a difficult team to play against. They make it very hard for any opposition they play so it will be a good test for us."