When I looked in detail at this issue about two years ago only the Iridium system offered coverage for the whole of Oz / NZ (SV, SMS would only work 10-50nm offshore and by his post it sounds like he needs a system to cover from Oz to NZ, a distance of approx 1,200NM).
Therefore a phone that operates on the Iridium network is probably what you are looking for. The Iridium 9555, 9505a, 9505 and 9500 will all work. I ended up buying a 2nd hand 9500 on E-Bay, however if I was to do it again I would probably look at a newer model.
I also believe some newer models have been released over the last 12 months and some of them have features that automatically switch between local Telstra services and satelite depending on coverage.
Note that sat plans are horribly expensive, expect to pay $3 per minute plus GST for most plans. I ended up purchasing a pre-plan 200 minute block as the cheapest option, however it depends on your usage.
Again you can rent, however I found it was far cheaper to purchase 2nd hand and then sell the phone and unused airtime.
I found pretty competitive. They have rental plans as well as straight airtime plans where you provide your own phone.