What a good idea. We are planning to go to the Azores in June/july, after the rally Portugal. Thought of buying an Iridium phone at the LBS, but the guy who runs the SSB course I am planning to take has warned me that it wouldn't necessarily be too good an idea to buy into that system as the satellites were designed for a 5 year life, seven years ago. This could mean that in a few years we would have expensive unusable phones. Seemingly the original company that set everything up went bankrupt, and the new company hasn't any resources to replace aging satellites, but hiring a phone seems to be a good compromise.
I bought a Satphone connected to the Globalstar network at last year's LBS for £99. I had to transfer to the Vodaphone network but the phone takes the SIM card from my normal GSM phone and defaults to GSM if there is GSM coverage. I have found Sat coverage to be very good, and calls cost £1/min. (That may be cheaper than GSM on International rates.) Only trouble is that this phone is first generation: battery life is limited and SMS is so complicated that you have to recourse to the manual every time.
Unless your Azores trip is a one-off it may be cheaper to buy rather than rent