As far as I could research, tuby sound was never the one to go after, because, in general, it doesnt have the high end part that many, who owns tube gear, include me, are seaking after. From my expirience, tubes can sound delicate, lovely, lots of air and detail, glorious midrange, but at the same time, and pretty often, without tighten, deep bass and real focus or structure. And this is what I am trying to combine in every new set up I build, which means lots of tube rolling, especially Input and Driver section, however depends on amplifier you are using, because not every tube amp have the same basics, where the caps, or even IC, are playing a huge part as well. Most of the time, its a combination all together, but back to the tubes part...
I would suggest to try some Mullards in your input and driver sections first and, of course, if you use an preamp, in your pre section to. The best results in the preamp section you may get from Telefunkens, which are known for their neutrality. Input and driver sections would benefit at most from very earlier, 1950-70's production years, Mullards, but I wouldnt go later than say 1968-69, however earlier are better.
For your output section I would try NOS GEC KT88. After I upgraded to them in my PrimaLuna Prologue Two, I just stop rolling output section. I dont believe you can find a better tubes than these. They are combining the power and delicacy, but the most are their ability to reproduce the music at the most natural way.
The only EL34 I tried were NOS TESLA's and they were very good, not as good as GEC, but probably one of the best EL34 up there. To me, they sounded bit laid back compared to KT88, KT88 are very powerfull and, if you get the right one, very musical tube and the same time.