I've rented sat phones for camping trips to Death Valley over Thanksgiving weekends. I used to rent Qualcomm's phone, but they used the Globalstar satellite system was apparently disrupted by radiation damage; check before you use a phone on Globalstar. Iridium continues to work fine.
See my delicious list of satphone companies at
http://delicious.com/philip/satphoneYou can rent by day, week, or month, but companies play with their shipping charges to make the costs hard to determine. Sometimes companies have a higher weekly rate by include free shipping, and the companies with lower rates charge you high shipping that makes the total cost higher than the higher weekly rate. If you get my drift. You also may find that the weekly rate is lower, but the per-minute charge is higher, so having an estimate of minutes in use is handy, maybe necessary depending on your budget constraints.
Death Valley has no trees, so we never had connection problems. I've made test calls in tree-shaded areas; since you don't really know where the satellite is, you can know whether a tree will block the call till you try.
There may be a delay when you stop and the person answers, like you see on those TV news broadcasts with the satellite delay.
Per-minute charges, last I checked, ranged from a dollar to two dollars.