Looking at the amps I've been through, it's two EL84 equipped amps that do real nice cleans at low volume: Budda Superdrive and Tiny Terror.
For the TT, the killer setting is Master and Tone at Max, set gain to taste for the volume (e.g. 8 AM).
Probably a Fender Champ is also a really good choice, or any 18W Marshall design.