Umaga, Blair and Neil Barnes were named on Tuesday along with Jason Ryan, the only assistant to former coach Scott Robertson to retain his job.
Blair played 85 Tests for Scotland from 2002-12 and is currently assisting Rennie at Kobe Steelers in Japan League One. Blair previously was a Scotland assistant coach and coached at Edinburgh and Glasgow Warriors.
Umaga is the head coach of Moana Pasifika in Super Rugby Pacific and previously coached Toulon in France and the Auckland-based Blues.
Barnes is head coach of Taranaki in New Zealand's National Provincial Championship and will be Rennie's senior assistant. He will work with Ryan who retains his role as forwards coach, while Umaga takes responsibility for defence and Blair for attack.
"This is a very experienced and highly skilled coaching team," Rennie said.
"Each of these men has a track record of getting the best out of the players and teams they coach and they are each hugely motivated to get stuck in and deliver the same success with the All Blacks.
"Neil provides outstanding leadership and strategic support, (Ryan) has proven himself as one of the best forwards coaches in the game, Tana brings immense mana and defensive insight, and Mike will bring his innovative approach and attention to detail to our attack. I know this group will challenge and support both me and our players to bring the best out of us all."
Former Robertson assistants Scott Hansen, Tamati Ellison and Bryn Evans were not retained, though Ellison will be the New Zealand Maori coach.
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