The TOS transporter no doubt features some redundancies somewhere, but those aren't evident from the technobabble. Save for the fact that there appear to be multiple transporter rooms aboard the ship, represented by a number of dissimilar sets, yet if one is down for reason X, generally all the transporters of the ship are considered to be down - but that was how it worked in TNG as well. It's just that TNG had an actual Tech Manual, plus dialogue stating the existence of multiple transporter rooms even when there was just one set that very seldom was altered in any way.
I'd suppose that if TNG has quadruple redundancy, the rough and ready TOS has at most a single level of it.
Then again, some episodes seem to feature a six-person maximum for a single transport sortie, while others do not, perhaps because redundancies in the system are sometimes utilized for safety, sometimes for capacity?