If you have to then approach your (if you have one) private people insurance like Southern Cross and see if you can get a discount if you join your dog and link it to your plan.

If I may be so bold as to suggest an alternative - Put aside $10 a week into a separate account from the very moment you get the animal and keep that as your 'pet insurance' money. 

To put things into perspective a surgery for a an amputation can be around $500-1000 and a surgery for implants if your animal gets hit by a car is around $1500-2500. (more for larger animals as anesthetic prices increase rapidly.) 
Ongoing dental care is usually around $200-300 a year depending on what you feed your animal (wet food = more dental) as they need to be anesthetized which is the really expensive part. 
And even euthanasia costs can be around $200-300.

Shop around for a good vet, there are many good ones and unfortunately some who are simply greedy ones who will charge you thousands where others will charge low hundreds for the exact same procedure. 

Source: My wife works for the SPCA :)

We got our puppy and two cats insured at the same place that we got the house, contents and car insured.