We don't hide an extra key. We also don't lock the door handle from the inside though, and our doors don't auto-lock. We are in the habit of having to manually lock them.
In your case, if I needed to, I'd do a coded lockbox (they lock over like a pipe or something sturdy) in the shed, if you leave that unlocked. Or even if you don't, I'd do a key to the shed somewhere outside and then the key to the house in the shed in the lockbox.