Telecom will announce compensation plans for "significantly affected customers" early next week, as it focuses on getting its XT network operational as quickly as possible.
Telecom chief executive Paul Reynolds said the "clear signal" from customers was they wanted the issue resolved.
"That has been our full focus today.
"Now we have a clear plan, the time is right to think about compensation."
Telecom spokesman Mark Watts said technicians were working in Dunedin last night and had restored sites in St Clair and Corstorphine.
Sites at Dunedin International Airport, Mosgiel, Otago Polytechnic and Halfway Bush were "high on the priority list".
As of yesterday, other areas with no or intermittent cellular service were in Invercargill, Timaru and Queenstown.
Service problems were still being encountered in parts of central Wellington, Taranaki and Ruapehu.
The fault affected voice, SMS and mobile broadband communication.
When Mr Watts was questioned about customers being able to break contracts with Telecom, he said it was part of the "broader discussion around compensation".
Earlier yesterday, Gen-I chief executive Chris Quin fronted up to about 200 customers in an online discussion, during which he was asked about the network's reliability and when it would be functional again.
Mr Quin conceded Gen-I would have to "front up on reliability", but that no network would ever be 100% reliable.
He could not say when there would be a return to full service.
Telecom developed the XT network in partnership with French company Alcatel Lucent.
Mr Quin said two executives had flown to New Zealand to assist, while technicians around the world were also helping.
Mr Quin said Telecom staff had been working through the night to narrow down the problem, linked to Christchurch's XT Mobile RNC switch.
Dr Reynolds has commissioned an independent review on the network and Mr Quin said recommendations from that review would be shared with clients.
