By the time of the War of the Ring, Sauron had retrieved and now held in his possession the Nine Rings of Men and (at least some of) the Seven Dwarven Rings. Since the Ringwraiths no longer possessed their rings yet still maintained great power, could he have re-gifted these 10-16 rings to more Men in the Third Age and created more Nazgûl? Or would it have just served to weaken those already enthralled to him? Could the surviving Dwarven rings could have been re-distributed toward the aim of creating new servants?
He could... but probably not very usefully.
Gandalf's explanation of the 'Ringwraithification' process (LotR - "The Shadow of the Past")
"A mortal, Frodo, who keeps one of the Great Rings, does not die, but he does not grow or obtain more life, he merely continues, until at last every minute is a weariness. And if he often uses the Ring to make himself invisible, he fades: he becomes in the end invisible permanently, and walks in the twilight under the eye of the Dark Power that rules the Rings."
Becoming a Ringwraith is just a consequence of a mortal using a Great Ring a lot, over a long period of time. Therefore, Sauron doesn't have to do anything special (or hold the One Ring) to make it happen — so it would work in the late Third Age just as well as in the Second.
Gollum isn't a counterexample to this because he didn't use it "often" for most of the time he owned it; he only wore it frequently at first.
The Hobbit, "Riddles in the Dark":
Gollum used to wear it at first, till it tired him; and then he kept it in a pouch next his skin, till it galled him; and now usually he hid it in a hole in the rock on his island, and was always going back to look at it.
So, if Sauron handed out the three remaining of the Seven (he still needed the Nine) to mortals (Men or Hobbits — Dwarves seem to be a separate thing), and they used them often enough for long enough, they'd eventually become new Ringwraiths.
That doesn't mean this would be useful for Sauron, though.
When he made the original Ringwraiths, he had the One Ring to control them, so he could eventually get their Rings back from them. Without the One, he'd have no way of preventing them from running off and setting up on their own.
Also, we don't know how long the process of becoming a Ringwraith takes: Sauron stole the Nine Rings centuries before the Nazgûl appeared. Given that Sauron had only declared himself openly less than 70 years before LotR, he might not have had time.
Now the Elves made many rings; but secretly Sauron made One Ring to rule all the others, and their power was bound up with it, to be subject wholly to it and to last only so long as it too should last. And much of the strength and will of Sauron passed into that One Ring; for the power of the Elven-rings was very great, and that which should govern them must be a thing of surpassing potency...
Of the Rings of Power and the Third Age
The conclusion seems plain: Sauron needs the One Ring in order to be able to control the others; given that it was no longer in his possession at the end of the Third Age he would therefore not have been able to distribute the other rings and create new Ringwraiths (at least not under his control).