In Germany (where Amazon uses DHL), DHL has "PackStations" scattered around the country where you can go to pick up your package if they miss you at home delivery. I think in most cases they're within half a mile of the home they tried to deliver to.
So when Amazon misses me, I just walk over with the DSL slip that says a package is there and scan the barcode on it and grab my package.
The Danish Post has installed such a system for general packages here. If they cannot deliver, they stick it in such a machine (open 24/7) and you can pick it up. But you can also put a special delivery address on your package so it will end up there right away (and you'll get a text message when it's there).
So when Amazon misses me, I just walk over with the DSL slip that says a package is there and scan the barcode on it and grab my package.