Spurs have struggled with hamstring injuries recently and have lost another man
Postecoglou added that his side have been working hard to sign reinforcements
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Timo Werner is the latest Tottenham player to be ruled out by a hamstring injury.
Ange Postecoglou has ruled the Germany forward, on loan from Leipzig, out for up to four weeks after scans revealed the damage.
Werner's form has been poor, but his absence deprives the Spurs boss of options and flexibility from the bench, just when he thought his injury crisis was starting to ease.
Richarlison and Mikey Moore have returned to the first team in the last week and defender Ben Davies is expected to be back for the game at Everton on Sunday.
But Werner is now sidelined and Brennan Johnson doubtful for the trip to Merseyside after picking up a slight calf problem in the 2-1 defeat at Arsenal on Wednesday.
'He's had a scan and he's got a hamstring tweak, so probably three to four weeks,' said Postecoglou on Werner.
'Whilst we've had a lot of injuries, we've also asked a lot of players during this period, and we've had to try and find ways to manage how much they play so we don't lose anyone else.
'We've sort of half got away with it I hope but even for guys like Brennan, he picked up a tweak off the back of even him trying to manage his load over the last two months. It has been a big shift for the players in there.'
Hamstring injuries have proved a particular problem for Tottenham this season.
Micky van de Ven, Wilson Odobert and Destiny Udogie are all ruled out at the moment with hamstring injuries.
Cristian Romero, Rodrigo Bentancur and Guglielmo Vicario are also out and will miss the Everton game with other injuries.
Postecoglou signed goalkeeper Antonin Kinsky this month and says the club have been working hard to find further reinforcements.
'It is tricky as all clubs are finding,' said the Spurs boss. 'I guess traditionally it ramps up in that final week so we're approaching there for all clubs. It's even trickier in January because once you try to acquire a player, the club wants to replace that player. So there's all sorts of logistics involved.
'It's not frustration. It would be frustration and disappointment if the club weren't working hard, but they are. It's an easy word to say but I know how hard the club have been working behind the scenes to get something done, but it's not easy.
'You need agreement from three different parties. Sometimes, when you're in our situation, other clubs realise that and try to make the most of it. That's why I was delighted to get Toni in fairly early.
'That's helped us for sure, but it's not for the want of trying so I accept that and I'm not disappointed or frustrated. I'm actually pleased at how hard the club is working.'