But there will be no "free data days" like those that followed service failures earlier this year, with Telstra instead offering customers discounts on their bills.

Chief Operations Officer Kate McKenzie said Telstra had run a software update on one of Telstra's domain name servers on Thursday, which caused that server to crash.

The fault was fixed by Friday morning, Ms McKenzie said, but Telstra did not anticipate a flow-on effect in which the crash reset customers' modems.

"A number of the modems sitting on the end of the network didn't behave in the way they were supposed to," she said.

"So since that time we've had to go through pretty much modem by modem, and domain by domain and customer by customer to figure out what their issues were."

Ms McKenzie said some customers - numbering in the "single-digit thousands" - had still not had their connections restored. Telstra is sending new modems out to those customers.

About 370,000 Telstra customers - or 10 per cent of all clients - lost internet access in the initial outage, with 33,000 still disconnected on Friday and 15,000 unable to log on over the weekend.

"We are incredibly apologetic to our customers for the inconvenience that's been caused to them," Ms McKenzie said.

"We hate doing this to our customers and we're going to work really, really hard on doing the best we can to be resilient and stable."

Ms McKenzie said Telstra continued to have "absolute confidence in our great network."

"We've invested in it substantially and continue to do so. It's one of the most advanced networks in the world."

She said the increasing complexity of networks meant routine updates were more likely to cause unexpected knock-on effects.

Telstra's network and customers' modems were increasingly complex, making routine updates more likely to have unexpected knock-on effects, Ms McKenzie said. However, she insisted the outage - the fourth major incident this year - was not caused by lack of investment in the network.

Asked whether the telco would offer affected customers another "free data day", as it did after previous outages, Ms McKenzie said customers had indicated they weren't interested. Some have been offered a $25 discount from their bills and Telstra is discussing compensation with business customers.

Ms McKenzie said she there was no sign of customers jumping ship to other providers in the outage's wake.

Telstra chief executive Andrew Penn had tweeted earlier this week that he was reading all their complaints on social media. 

To customers still experiencing difficulties with your broadband service, pls know I'm reading all your comments and our team is on it— Andrew Penn (@andy_penn) May 24, 2016