Yesterday's "high demand" from O2 customers had also led to further issues with voice calls.
Pay as you go mobile broadband customers will get a 10% discount on a bolt on purchase in the new year - 02 will let notify when it is available.
O2 blamed the network issues on a third-supplier and Ericsson, with the company issuing an apology to its customers as it worked to restore connection.
The CEO of O2 Mark Evans followed it up with: "I want to reassure our customers that we are doing everything we can to fix the issue with our network and say how sorry I am to everyone affected".
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Pay-monthly customers, SMB business customers and mobile broadband customers two days of monthly airtime credit off their bill by the end of January. After conducting a root analysis, the company said that "the main issue was an expired certificate in the software versions installed with these customers".
The day-long O2 outage impacted all of the operator's 25 million customers, as well as an additional seven million customers using Sky Mobile, Tesco Mobile and giffgaff, all of which use the same mobile infrastructure.
The network provider worked with engineers from telecommunications company, Ericsson, to resolve the problem which had affected millions of its customers.
In total the Ericsson software glitch cause issues for carriers in 11 countries. We are really sorry and working as hard and as fast as we can to fix this.
"We'd like to thank our customers for their patience during the loss of service on Thursday 6 December and we're sorry for any impact the issue may have caused".
"We have been working hard on resolving the United Kingdom data issue since early this morning", said Lindgren. "Connectivity is now such an integral part of our lives, it is time for the regulator to consider whether it should introduce automatic compensation for the inconvenience caused by severe outages", he added.