Optus chief executive Kelly Bayer Rosmarin said customers had been let down and it was integral for them to be connected in the modern world.
"We really appreciate the patience and understanding that our customers have shown," she told AAP.
"We know that there's nothing we can do to change what happened yesterday, but we are going to be giving this offer to our customers to thank them for their patience and their loyalty."
Customers will have access to 200GB of extra data for free over the next few months, which Ms Rosmarin says will provide freedom to "use the network, and enjoy all of their streaming, surfing and other services through the holiday period".
Small businesses disruted after the nationwide outage ground trading to a halt will also be able to take advantage of the deal.
More than 10 million customers and businesses were thrown into disarray from about 4am on Wednesday, with customers prevented from connecting to the internet or making or receiving calls.
Ms Rosmarin said the outage was the result of a "technical network issue" after it experienced a "network event" that triggered a cascading failure.
The telco's extensive levels of fallback and redundancy meant it had to undergo a complex diagnosis, which meant it took longer to resolve.
The Greens on Thursday secured a senate inquiry into the outage, to be chaired by the party's manager of business in the upper house Sarah Hanson-Young.
She said the public deserved better.
"We want those affected to be fairly compensated ... so this doesn't happen again," Senator Hanson-Young said.
"The inquiry will look at what responsibility Optus has to protect the public, not just their profits."
The Greens will seek to set up the inquiry as soon as possible.
The federal government will launch a review into the network failure.
Michelle Rowland says it's critical to conduct a process to identify lessons from the Optus outage.
Communications Minister Michelle Rowland said connectivity was "absolutely essential" for consumers and businesses.
"While we welcome that Optus services were restored over the course of the day, it is critical the government conducts a process to identify lessons to be learned from yesterday's outage," she said on Thursday.
The minister said her department would develop the terms of reference for a review.
The Australian Communications and Media Authority has independently started to investigate Optus' compliance with rules on emergency calls.
The telecommunications watchdog is urging small businesses that were impacted to raise compensation with the provider.
"What we would encourage you to do is contact Optus and ... help them understand what the impact was on them and their earnings," Telecommunications Industry Ombudsman Cynthia Gebert said.
Ms Gebert said the scale of the outage was unacceptable as it impacted emergency services and hospitals in addition to small businesses and transport services.
Optus chief Kelly Bayer Rosmarin has "no idea" how long it will take to fully review the outage.
It took more than 12 hours to restore services.
Businesses had been dealt multiple blows, a day after the Reserve Bank increased interest rates, National Retail Association director Rob Godwin said.
"This is costing businesses thousands of dollars in sales that they are now in dire need of given yesterday's rate hike," he said.
Victorian Premier Jacinta Allan said the government would review the state's contracts with Optus "as well as the telco's response".
Almost 10 million Optus customers had their personal information stolen during a massive data breach last year.
People weren't able to call triple zero on landline devices, although it was still possible to do so on a mobile phone.
