You can use KT-66 tubes in a Torii MK III. Several here have done it and like the result but it doesn't utilize the Hazen Grid Technology. Below is what Steve says on page 4 of the owners manual...link provided.
I've got a MK IV and except for a brief time running the stock KT-66s I ordered with it, I've been using Seimens/RFT EL-34s. I like these best but I seem to be in the minority on this. Mark.
Unlike the TORII MK II that used EL34, KT77, 6550 or KT88 without re-biasing the TORII
MK III uses our new Hazen Grid Technology to push the transparency, speed, tightness to
a level that greatly surpasses the alternate tubes. Since this is the case, extra voicing
was used to get the maximum sonics from the EL34 at the cost of making any of the
above listed alternates a waste of time because they will not sound as good.
https://www.decware.com/newsite/TORIIMK3Manual.pdf