My answered is based on TNG-era transporter technology, but at the moment, I can find no evidence that the particular features I am about to describe are different in the TOS era.
According to the canon Star Trek: The Next Generation Technical Manual (written by Michael Okuda and Rick Sternbach, production designers for TNG),
Since the transport beam conduits permit the matter stream to be routed to any pattern buffer, any given chamber can be reused immediately without waiting for cooldown by switching to another pattern buffer. Since there are only three patter buffers normally used for personnel transport, this process can be repeated twice before waiting for pattern buffer reset. This translates into an average of about 1.9 six-person transports per minute...
(From Chapter 9 "Transporter Systems", Section 9.4 "Limitations of Use")
So it seems that transportation is not related to the number of pads or even the position of the pads. The transport beam conduit can be directed into any pattern buffer. Rather, the key is the number of independent pattern buffers.
It seems that the six pads are simply a suggestion, corresponding to the average situation presented above.
This is consistent with the fact that we see many occasions when transport is accomplished without pads at all, using targeting scanners only (e.g. transport from a surface to the ship or site-to-site transport).
There does seem to be an upper limit:
...six emergency transporters capable of transporting twenty-two people at a time.
(From Section 9.5, "Transporter Evacuation")
The emergency transporters have a higher capacity than usual transporters (at a trade-off with distance) and so a normal transporter will have an upper limit of less than 22 people.