I looked at the NPS listings, and I was surprised that there were not more possibilities than there were. In fact, when you rule out infantry privates and non-coms, that leaves very few possibilities for this person.

By my count, there are 8 possibilities. [Note that the NPS listings, however, do not include Virginia listings, yet.]

And I highly suspect that the Capt. T.W. Richards shown in three different Mississippi units, may be the same person.
And, the Thomas Richards of the 3rd Ky Cav is most likely the same as the T. Richards of the 1st Ky Cav, as the 1st was merged into the 3rd.

If the above speculation is proven out by the CSR's, you are down to only 5 possibilities. For whatever it may be worth, of those 5, only 1 is in a Tennessee unit.