Boon:
With you current system you can try an RCA Black plate or a Sylvania Gold Brand 5687 (both offer a full sound with the GB Sylvania being more refined). I used and enjoyed both types (depending on my mood) with an Audion Silver Night Stereo 300B.
Given your goal, spend the money for a decent testing 1960's Mullard ecc88 or e88cc tube for the input position. Amperex will not have enough weight, considering your complaints with the setup, otherwise a vintage Amperex (white or orange label) would be a good choice - with softer/fuller sounding speakers.
The Svetlana EL34's are a good pick, IMO, considering your musical tastes.
I paired Reynaud Twins with the Silver Night for a year, or so, and it was a very good combo. Perhaps the latest (more efficient version) would be a decent match for the Sterling?
The amp is a good one and I feel that you will be happier (again with you musical tastes) building your system around it.