Global Rescue is very highly regarded by most who belong. I will be using them the next time I hunt in Africa, without hesitation. Certainly for dangerous game hunting, but I know a fellow who was hunting in Mozambique who used GR to evacuate him to Johannesburg, as he had an eye problem. Turned out, he had a detached retina and was immediately flown back to the USA for treatment. Pretty cheap insurance if you ever need it.
You shouldn't need a satellite phone in South Africa, as they have excellent cell phone coverage. Advise your carrier in advance that you'll be in South Africa and you shouldn't have any problems. I was in Kruger Park a few years ago, and talking on my ATT phone with a client here in California. She had no idea that I wasn't in the office until I told my son to come see the buffalo that were coming into the waterhole, which was lit up behind our guest house in Satara Camp.