Only if you never cut off the scapes, and the scape matures, and splits open. Not sure how you would get garlic you didn't want though.

I do not believe your barber is on the cutting edge of allium culture. He seems to be confusing garlic with garlic chives. The latter can become an extremely invasive pest due to their wondrous self seeding ability. It takes years to erase their growth unless kept under strict control by plucking it out. They are a real bother and not a case of hair today and gone tomorrrow!
Garlic in zone 6 is a wll behaved annual as you have accurately researched. It is not a difficult root crop. One might say it is easy to be a garlic razor and its taste is a cut above other alliums. You might want to wave off your hair cutter when it comes to gardening advice.

Actually, there is a place above my garden where some rocambole garlic was grown ten or fifteen years ago. A few plants survived when the space went fallow, and they have multiplied into a feral patch. The plants are quite crowded because they grow from whole bulbs, but in early spring I dig them and use them as g'allions -- garlic scallions.

There are many patches of wild elephant garlic in Zone 7/southeast, which probably grew from topsets.

On the other hand, when you harvest at the right time, you will never miss a bulb.

I not sure one can have to much Garlic.
There are 50 or so varieties.

Thanks folks! Now I'll no longer have dreams of a giant garlic bulb chasing me through my garden! I wanted to disagree with my barber, but he was holding those clippers....Don't know how I might look in a Mohawk....... ;)

So.... OP is a vampire? ;)

Yuuup! With a Mohawk.......

Garlic is an excellent and vigorous replicator, so I would say the barber is correct if that is what he meant. Much like walking-onion, garlic propagates from top sets and from the basal cloves. Very prolific.

Soooooo, there seems to be a difference of opinion in this thread........ What I'm extracting from this is, even if you dig all the bulbs from the previously planted garlic, the flowering scapes will reseed, thus producing volunteer plants.

Garlic scapes should be removed before the bulb reaches maturity. In about thirty years of growing garlic here in Massachusetts, I have never had one reseed.
Garlic chives are another plant and another matter as previously stated.

It's pretty easy to miss some clumps while harvesting and those come up in early spring as big clumps of garlic scallions, as mentioned earlier.

OK........ Now I think I've got it..... Removing the scapes prevents reseeding. Obviously my barber didn't get the memo. ;)If I miss one I'll have g'allions......

Never argue with your barber or your proctologist. You have more to lose than they do.