Gift bags are handy for weird shaped presents and a quick wrap job. They are also hard to contain neatly.
The most efficient and neat way is to get a hanging gift bag organizer This is similar to the hanging gift wrap organizer. It has different size pockets for various size bags and tissue paper etc. It is also double-sided.
It's really the only good way I've found for organizing gift bags. You can always put the bags inside of bigger bags or in a box but you have to dig though them to find the right size and occasion bag.
