Try the Seimans 7308 if you are looking for warmer than the 6922 but still neutral. Or for extremely warm try the Amperex 7308. Both very tough to find.
Two other choices are the Mullard gold pin 6922 and Telefunken 6DJ8. The Tele is the most beautiful tonal balance, and by far the most expensive. The Mullard is not as difficult to find, and could be the least expensive of this group. I have a wholesale catalog that lists them NOS, in original box at $24.90. I've purchased about 20 of them so far, and had only one bad among the group.
The Mullard is what I am running as inputs for my Wolcotts, and in this particular circuit, they sound better than everything else suggested here. My Aesthetix gear plays the best with Telefunken, and reacts poorly with the same Mullards that works so well for the Wolcott.
If you are in tubes for the long term, all of these reference tubes are a necessity. If you can afford to get a few pieces of each, you will not regret it. Tubes that are not perfect for the BAT right now, could be first choice in a future piece. Should you decide to return to transistors, you can sell off your inventory at a profit, as prices are continually increasing. Following this route, you will discover the ideal tube for the BAT with your current gear, and have spares to test with, should you change components.