I don't know how I missed this thread until now...
A stock Stax amp is fine for most users but as you edge more and more towards the fringe they aren't all that good any more. The tubes are easy to roll and I find the Japanese Toshiba to be the best match and the various USA made tubes to be a distant second. The amp simply snaps into focus with the Japanese tubes as it was initially designed with them in mind. You can upgrade the various parts of the amp but I would much rather sell the amp and buy either a Woo Audio GES or a Headamp KGSS. Both will be a clear upgrade over the Stax amps.
Other components such as source, cables and isolation devices are important but I recommend a solid foundation for the amp, no power filter/conditioner and silver wires throughout. The problem with silver is that it is very expensive to make it right so a lot of cables use badly made wires and thus sound horrible. The basic rule with Stax is to use something that is very transparent and natural. Power cables are important but since there is a simple linear supply in the Stax amps you need only a simple, well made power cord. No fancy emi/rfi shielding, just quality wire and connectors.
The cable on the phones can't be upgraded easily but since we are dealing with high voltage the cable behaves in a different fashion to a normal headphone cable. Capacitance is the most important aspect and the cable used on the SR-404 is the best that is currently available.
There are some Stax setups in your region and some of the owners might allow you to come and test their setups. You could check the meet section and see if a Texas meeting is on the books.