Depending on how much rain you get, a rain chain might not be efficient. You might end up with a more condensed torrent, dropping at a fast rate into one very small area. That will cause a whole host of problems like dirt splashing up, a crater being formed, worst case scenario a minor flood. It also won't really accomplish what you're hoping as far as watering the garden because it will be localized to that back corner rather than distributed across the whole bed.
I'm a big fan of rain barrels. What if your rain chain (or downspout if your climate requires) ended in a rain barrel in the corner there? Sometimes people elevate them on platforms and use a spigot for easy filling of watering cans. This is a pretty example, but not raised.
Q