So. I’m based in Canada so I can’t help you with actually renting one but I’ve used 3 different systems of sat phones. Basically the more you pay the better the service. So in order of increasing cost / decreasing quality…
Imarsat – This is designed to work for boats as the satellites are geo-stationary on the equator. If you can see South (exact angle varies) and you pick up the satellite you’re fine but otherwise it won’t work. So great if you’re thinking beaches etc; less good if you’re planning to climb the North Face of a Munro
Globestar – Not geostationary satellites but not enough to cover the world but does work in the UK (I think). These are good and work well most of the time. In mountains have had occasional dropped calls when one satellite leaves sightline and another hasn’t yet made it over the horizon. They are the signal carrier for spot.
Iridium – The gold standard and has always worked but pricy.
10 seconds on google found this http://www.satphoneco.com/%5B/url%5D
http://www.satphoneco.com/rentals