Garlic doesn't produce real seed, but hardneck garlics do put up a stalk that has top-sets on it, called bulbils. I plant some bulbils every fall, along with some cloves from mature garlic bulbs, but only to increase my stock. The bulbils will produce very small "rounds" the next summer, not the "spring garlic" you are looking for.

So here's what you want to do. To get mature garlic bulbs, plant the garlic cloves in the fall. I plant mine in October, then pile the bed with chopped leaves and cover the whole thing with a layer of pine straw to hold the leaves down in windy weather. Harvest the next June or July when about half of the leaves turn yellow. To get Spring Garlic, plant the cloves (from mature bulbs) in the early spring when you plant lettuce, etc. Add a layer of composted manure (onions and garlic like nitrogen). Harvest when the leaves get big enough to look interesting, but certainly by the time the leaves start to yellow. Eat the bulb, or replant the rounds in the fall for a mature crop.

Hope that helps.

Catherine