That first statement needs a link to some official source. Where does it say on the INZ website that anyone can stay in NZ for one and a half months with no visa?
if you have a new offer, one can apply for variation visa
.While you are on a valid visa, you can apply for a variation of conditions, if you meet the regulations. But as I said above and linked to the regulation, a person may not apply in the normal way if their visa has expired. And INZ does not have to grant another visa if they ask for it under Section 61.