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I have 'California White'
I have 'California White' soft neck and 'Music' hard neck that came from grocery stores more than 20 years ago. They took a few years to develop, but they eventually adapted and have grown well for me in different parts of the US. The concern of store bought garlic is that it is not seed grade and may be carrying some sort of pathogen, If I was you, I'd try all varieties available and test grow them in plots that are separated just in case there's a problem with one type. It will take a few years to determine how well each will perform in your garden. Your locally available garlic that is small may be due to planting time or from not having enough time to develop.