Police now believe a third stolen vehicle, a silver Mazda registration FYT67 was also used in the aggravated robbery.
The White Mazda Atenza sustained a puncture in the front right tyre before the incident.
The offenders then changed that tyre somewhere in the Hillcrest area of Hamilton. It was replaced with a space saver, Ross said.
The original tyre was discarded by the offenders but has not yet been recovered by police.
Police are appealing for anyone who may have seen people changing the tyre of a white Mazda Atenza in this area to contact Detective Bernie Kelly at Hamilton Police on 07 858 6200.
"We'd also like to hear from anyone who may have seen the three vehicles, travelling separately or together, in the wider Hillcrest and Karapiro areas early on Monday morning," Ross said.
Information can also be reported anonymously to Crimestoppers on 0800 555 111.