You have some options as to what to plant for green garlic - you can plant a full head, individual cloves and just space them closely or the little bulbils that come out of the seed heads of hardneck garlic. All will work fine to get essentially garlic scallions. Garlic chives are a different thing - allium tuberosum versus garlic which is allium sativum - but garlic chives do have a very mild garlic flavor and work well in cooking though they are a bit invasive via seeds when left in the ground.

Depending on the weather in your zone, you could probably get green garlic for harvest in either spring or fall. The problem with succession planting for full season harvest is that garlic is day length sensitive and it will drop all its leaves and go dormant during the long hot days of summer. In zone 5b, if you planted for green garlic now, you might get some harvest but by mid July almost all of my garlic has died down and bulbs left in the ground don't start again until mid September at the earliest.

Thank you! This is a great answer, and very helpful. I've posted this question several places and no one has seemed to know the answer, so I'm very glad to have your response.