Keep those boxes. I use them to store clean empty jars in my garage, and I also use them to store full jars on my kitchen pantry shelves as well as for my additional storage. They keep the jars together, and keep them from getting tipped over. I can mark the boxes on the outside to know what's in them.
I use a "ring hanger" - two wire hangers twisted together - to store extra rings in addition to a couple boxes full. I like a heavy plastic bag with a reinforced handle area to cover them but make them readily accessible. I usually hang it on my jar storage rack or inside my pantry door. Be sure to remove the rings from your thoroughly cooled, sealed jars and carefully wash the jars, especially in the screw band area. This will prevent mold and rusting.
Make sure your shelving is STURDY. Filled and even empty jars weigh a lot, and a collapsed shelf will certainly make a mess!
I have also stored canned goods in the boxes under our bed or in other closets when I haven't had as much room. I keep a list of what is where.