Prop Ethan de Groot and lock Josh Lord will both undergo concussion protocol testing after their heads collided and they were replaced in the first half of the All Blacks' 33-16 win in Johannesburg.
A New Zealand Rugby statement said experienced Auckland Blues prop Tu'ungafasi and Canterbury Crusaders lock Hannah were set to join the tourists on Monday.
The pair could be swung straight into action to play in the final midweek tour match, against the Lions at Ellis Park on Tuesday.
Six additional players in total have been added to New Zealand's original squad of 44 because of injuries.
There is uncertainty over the availability of back-rower Anton Segner for the remainder of the tour, which comprises the Lions match followed by Tests in Cape Town, Johannesburg and Baltimore in the United States.
German-born Segner was named on the replacements bench for the first Test but was withdrawn hours out from kickoff with a swollen ankle.
No time frame has been put on his recovery.