Without having seen your company constitution (which may limit the activities the company can engage in), as a matter of general law you wouldn’t need to set up a separate company to do this.
You may however want to think about whether you want to set up separate companies so that the liabilities of one business don’t affect the other, and you would also need to talk to your insurer to confirm whether operating these 2 types of business would be permitted under your insurance.
If you are in leased premises, it is also possible that your lease limits the activities that you can conduct from those premises.
You also might want to check out the NZTA website or call NZTA to check whether your rental operator license prevents you operating as driving instructors, or vice versa.
Hope this helps