It sounds like you have a top-setting garlic that produces flowers, then bulbs at the top of the plant. The stalk as it matures bends over and releases the babies you see on the soil surface. They become the next crop.

Fall is the time when folks plant garlic. If you wanted to, you could scratch them into the surface a bit. As to the larger bulbs under the plant, they may be quite usable even though it's October.

Your "walking garlic" sounds like an heirloom considering that it's been growing for many years. How nice! Maureen

Thanks for the information. You described it perfectly. I must have an heirloom. It has been there for 40 years. Plus, I do not think anyone including my mother ever dug any up. So, when this dies down in the spring I did it up?
I think the stalks have big round flowers on top if my memory is right on this.

No, don't dig it in spring. You can dig it now and space the larger cloves out in a row, say 6-8" apart. Put plenty of compost in the row.
Then harvest next summer after the leaves die down but while there are still one or two green leaves. That way the bulbs will have tight wrappers.
Any extra cloves you dig up now will keep for a while and can be used in the kitchen.

Considering your southern location and the round flower tops it sounds like elephant garlic to me. Elephant forms huge round lavender flowers. Most garlic in the south that continues to come back is Elephant. They don't top set but do love to come back year after year and will multiply. We found where ever we plant elephant it will continue to come up and anywhere the little "corms" fall off the roots you will get lots of little ones. You can leave them in the ground now and will need to harvest in June. We had a volunteer come up in our field last fall . I left it where it was and it ended up being 1 .5 lbs at harvest!