Any way to reduce pre-amp noise floor?

If the noise increases with the volume control, then that at least rules out the preamp's buffer output stage, and your amp.

In a review of the ET3-SE, it's mentioned that this unit uses a single 6922 tube for voltage gain. They didn't mention whether that tube comes before or after the volume control. If before - then it would be a prime suspect as the noise contributor. You could buy a guaranteed "low noise" 6922 tube from a trusted tube vendor, and see if that helps. If your source is an MC phono stage, then that could also be the (likely) cause of noise.

If all gear is in spec and the tube in question is not a noisy specimen, then it may just be that you've got an exceptionally high sensitivity amp & speakers - which would imply that you'd have to listen with the volume control quite low (i.e. you're throwing away gobs of gain - a prime candidate for a passive, though I still love the sound a good tube preamp even with some excess gain and noise floor).

BTW, being a vinyl and tube 'phile - I don't find a modest noise floor of the white noise variety (i.e. NOT ground loop hum) to be of detriment to the music.

Thanks for clarifying the volume-control/noise behavior. Clearly the unit is muting the output at volume=0.

I just looked up its specs: 25 dB of gain - holy COW that's a lot of gain (average these days seems to run ~ 11dB-14dB)! You're going to hear some noise floor with a linestage of such high gain, unless your amp/speakers are very insensitive (and thus would need all of that gain). Since it seems like the volume control comes before the tube gain stage, you're effectively "throwing away" gobs of gain without reducing the noise floor generated by it (I'm assuming you're keeping the volume control relatively low for listening). That's why your noise floor is discernible. I've got the same situation in my own setup (tube linestage, > 20 dB gain, 93dB/Watt speakers and 250-Watts@2V amps) - it's not detrimental to the sound quality in any meaningful way.

If this continues to bother you, you may consider sourcing some fixed attenuators to insert between your preamp & amp (i.e. attenuate AFTER the excessive gain stage). You'll effectively increase your signal/noise ratio by the amount of attenuation (maybe try 10dB). Just don't go too far to where you're gain starved on certain recordings.

Perhaps the gain is high so the 54dB phono stage can be used with a low-MC cartridge, with the extra linestage gain to cover the deficit. And I guess some manufacturers also try to accommodate for users with super-inefficient speakers.