Justice Minister Andrew Little says further suppression breaches by Google in the Grace Millane case are "disturbing" and "disappointing".
Google earlier apologised after publishing the name of the man arrested and charged with murder for the death of the British backpacker, despite name supression applying to the case. The name appeared in the company's 'what's trending' email in New Zealand last December.
It spurred Mr Little to consider setting up a database for court suppression orders to help media and technology companies avoid disseminating suppressed information.
Last week, Sky News Australia found another breach by Google. It was swiftly removed after the company was notified.
However, after Google had taken action, another breach was found by 1 NEWS on September 6.
"That's really disturbing," Mr Little said.
"We've been in touch with Google as recently as in the last couple of weeks. They said they would take down any references to the defendant in the case about Grace Millane.
"I'm really disappointed."
Ross Young of Google NZ said the company understood "the sensitivity around this issue".
"We have taken action, including suspending Google Trends emails about searches trending in New Zealand and removing reported web pages from New Zealand search results," he said.
"When Google receives suppression orders, we review and respond appropriately.
"We recognise this is an issue that needs to be addressed across industry. We acknowledge that we don't always get it right and look forward to working with the New Zealand Government and law enforcement on this important issue."
National's justice spokesperson Mark Mitchell told 1 NEWS there needed to be a conversation around "how we control what people are doing in other countries that breach our New Zealand laws".
"This is a very complex challenge that requires international coordination and cooperation between countries and social media companies in order to be able to respond effectively.
"The most preferred solution would be to see media outlets and social media companies come to some sort of agreement around a code of conduct themselves, but I would be happy to work with Minister Little to address this challenge."