I too want that spound but a tad louder.
If you put a current sensing resistor on the cathodes of the 6V6s, you can come up with a cleverly designed circuit to force a multiple of the same current through a pair of 6L6s, EL34s, or even 6550s. Or a quad. Or cascodes of MOSFETs. All you need is sand and feedback.
Of course, a quad of 6V6s is easier, you won't risk being commited by loved ones, and you already have the OT.
Don't expect much more clean headroom from a quad. You can raise your plate voltage to at least 420V, which might help, if you're not there yet. The Gm on 6V6s is really low, so you may need to ask more of your PI. It may be causing the lack of headroom that you are hearing.