Kirsten, is it possible to make a temporary second door inside the first one? We did this at our old place in the barn because our cats were always trying to escape as we opened the door to get in. I know it would be unattractive but you you could create something like this using plywood to create a space between the door and the rest of the house. This would allow your pet sitters to open the door, close it behind them, take a few steps and then slide the plywood out of the way. Or instead, you could use some plastic sheeting or tarps and nail them to the ceiling and put rocks or something heavy to keep them from getting in underneath. It would just give your pet sitter that extra 30 seconds to get in and out safely without potential 'escapees' on her tail.
I know this would look really strange but if it gives you peace of mind while you are away, trying to heal yourself, then it would be worth it.
Also, maybe you could have some tags made up before you go with your pet sitters phone number on them. If a cat gets out, someone will read the tag and call the petsitter.
Just some thoughts. Hope they help.