The telco has been in damage control since the outage last week, which left thousands of Australians unable to make calls or access data.
An EFTPOS payment system used by about 80,000 businesses confirmed it had been impacted, with a number of customers taken offline and unable to process payments.
Train services had to be cancelled after severe network disruptions in both Victoria and NSW, triggering strong criticism from the federal government.
Telstra chief Vicki Brady will be among a group of executives who will front a parliamentary inquiry into the incident in Canberra on Friday.
Greens communication spokesperson Sarah Hanson-Young said the committee holding the probe had called an emergency hearing over the outage.
"The truth is, Telstra, just like Optus, has put their profits ahead of public safety and public service for far too long, and the law allows them to," she told reporters at Parliament House on Thursday.
"We need better laws in place, stronger laws that protect the rights of the public, the rights of the consumer, and to force these companies to actually deliver a reliable service."
Telstra is accepting compensation claims from affected customers and small businesses who can provide evidence to support their case.
Senator Hanson-Young slammed the telco for putting the onus on impacted Australians.
"It is appalling that Telstra has done the bare minimum when it comes to compensation for consumers," she said.
"They should not be making Australians jump through hoops because they stuffed up.Â
"The company should be taking responsibility, and it should be offering automatic compensation to everyone who they put in a difficult and dangerous position."
Telstra undertook more than 630 welfare checks on customers that were prevented from having triple zero calls connected due to the outage.
In a statement published earlier this week, Ms Brady apologised for the failure.
"Last week we let you down, and I am sorry for that," it reads.
"When something goes wrong, we're committed to taking accountability, giving you clear information, and fixing issues as quickly as possible.
"We understand what caused the issue and will complete a full investigation and take what we learn to make our network stronger. You have my commitment on that."
Representatives from the Australian Communications and Media Authority and the communications department will also give evidence at the inquiry.
Telstra was contacted for comment.
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