My mother used to peel and chop up all her garlic put it in a canning jar and cover the top with olive oil - which makes a seal, and then put on the lid, and into the fridge, I don't think she canned it. That helps it keep longer than if you'd just let the bulb sit on your counter. when you need the garlic you've jar'd just use a spoon or measure to reach down past the oil to scoop out the garlic.
Here in FL the only garlic that comes in the mesh, are the garlic braids, I'm not sure if you could tie it down small enough to work on the spools, but I'll try it the next time I buy a garlic braid! :D That transparent thread never comes of the spool properly!