How do other auras interact with Gift of Immortality?
After a creature dies, but before triggered abilities are placed on the stack, State Based Actions are performed. The SBAs will cause both Rancor and Gift of Immortality to go the graveyard because they are not attached to anything [CR 704.5n]. 

Then, it's time to place on the stack the abilities that have triggered. Gift of Immortality's ability and Rancor's ability are placed on the stack in whatever order you choose.

When Rancor's ability resolves, it returns to your hand. When Gift of Immortality's ability resolves, it returns the creature to the battlefield and creates a delayed triggered ability, which puts Gift of Immortality back on the battlefield, attached to the creature at the beginning of the next end step.

Angelic Destiny's ability works the same way Rancor's does, so it would also return to your hand.