Putting aside the power supply issue for a moment, you have picked a low-noise opamp, but your schematic is sort of a noise fest.
The AD8676 has 2.8nV/√Hz input-referred voltage noise, and 0.3pA/√Hz current noise. Now you supply your virtual ground to the positive opamp inputs with voltage dividers and no smoothing at all. That means you get all the supply noise, and 5kohm worth of thermal noise which is about 10nV/√Hz.
Admittedly the input transformer (apart from capacitive coupling) turns this into common-mode noise on the first opamp, making it pass through to the output merely buffered instead of amplified.
What else? The negative input of the first opamp sees an additional 9k of impedance good for 17nv/Hz. If you put the volume wiper in the middle, the right opamp gets to see 35k on its negative input, good for about 26nV/√Hz voltage noise of its own, plus about 10nV/√Hz of noise from the opamp current noise, bumping this to 28nV/√Hz (uncorrelated noise power can be added, so for voltages you take the root of the added squares).
With the first input stage amplifying 5-fold (in middle position), this is an additional 6nV/√Hz when referred to the first input of your amp.
All that adds up. Your virtual grounds can just be given a capacitor for smoothing. You can share the same ground between both stages; that way the ground noise from the first stage is passed through the second stage unamplified (thus becoming irrelevant) iff your volume pot also takes this virtual ground rather than the real ground.
And it really should, or you have 1.65V DC across the volume pot which makes for very scratchy volume adjustments.