Here in Italy, we are often obliged to buy three or four garlic cloves together and they sprout so quickly even if you use a garlic jar. I usually have to throw out the last ones.
Now I divide up all the cloves and freeze them directly, without peeling, in a small ziplog bag. This way there is no wastage: the frozen cloves peel more easily than the fresh ones. They do not become stone hard so can be be sliced almost immediately.
If you use a garlic squeezer, even better from frozen - they become soft enough to squeeze easily (the fresh ones are very painful to squeeze with my arthriticky hands!)
Ps: remember that if you do use a sprouted clove you should remove the green interior. The name for this in Italian is "anima" (soul)!