Hi Grannyring - sounds like you've got the stock tubes so almost anything you do will be an improvement. Many of the Supratek users on the board here use the Tung Sol 5881 in as regulator tubes and Mullard GZ-30-somethings as a rectifier. That is a very nice combo but I'd be willing to bet you can find something you like better - experimentation is the key to finding what fits your taste and synergizes best with your system.
One tube you left out is the 6SN7 tube.... That's the KEY to the Supratek sonics. I recommend Raytheon VT-231, Tung Sol Round Plate 6SN7GT and Ken Rad Black Glass VT-231's as great choices for the Syrah. I'd say the Raytheon is the least colored followed by the Tung Sol and then the Ken Rad.
Sylvania VT-231's and the 6SN7WGT (metal base) are good choices for the most open sound you can get out of the Syrah. The 6SN7WGT's are expensive and sound like all sylvania's sound slightly metallic to me but if you're into music like Enya, there is nothing better.
My favorite 6SN7 is the Ken Rad Black Glass VT231.
More on Rectifiers:
The Bendix 6106 is great. Good strong bass... nice mids, nice highs. Overall, I'd say this is the best choice for use in the Supratek but.. there are situations where the Bendix will absolutely not work. Some tube combos just clash with the Bendix and the pace / timing of the music goes crazy.
The old GEC/MO-Valve/Cossor/Genalex/etc U54 is nice. Very nice! These are pretty rare.
Regulator tubes:
Any 6L6 variant will work and they all sound tremendously different. The best of the best are the GEC/Genalex KT66 if you want the ultra seductive liquid mids.. The WE or NU 350B for more brutal honesty - superb bass - a real "you are there" type of sound. Almost too much for me but its what I'm using now.
The Sylvania 5932 - three types - and they are all very good. The most common, black plate with holes variety is the most neutral and/or accurate / realistic sounding tube I've heard in this application except... neutral is boring.
The rare dual black plate version is almost as good as the GEC KT66 for liquidity and seductive sound but mixes in some of that accuracy for a nice blend. The black plate without holes version - I haven't spent too much time with but as far as I can tell, its similar to the "holed" version but offers a little more personality, character and as such is not as dull as the neutral holed version.
RCA Metal 6L6. Cheap & ugly. Wide soundstage but a bit thin & somewhat see-through.
As for the Syrah phono tubes - I do not have any experience rolling those so I cannot comment.
Ecclectique has in the past made some awesome "tube related" posts to this thread. Scan through his posts to get more insight on tubes.