Preamp Output Completely Different with Tube Amplifier versus Solid State Amplifier

It could just be that your two tube amps have higher gain than the First Watt and so amplify that tube rush more.

By looking at the specs, the First Watt has an input sensitivity of 530mV for a 1-watt output at 8 ohms and the ATM-300 has an input sensitivity of 290mV for a 9-watt output at 8 ohms. That is 29dB gain vs a 15dB gain for the First Watt.

Also, I have always had a problem finding quiet 12AU7 tubes. Inherently noisy tube plus high gain is most likely the problem.

Exactly. I have a First Watt M2 that also has 15db of gain. My tube preamp is dead quiet with that but exhibits noise with my 1V sensitivity tube monoblocs.

This is a consequence of system gain structure. You have a very high sensitivity speaker, with a lot of combined gain from your preamp and power amp AFTER the preamp’s volume control. In net, you have way more gain than you need, and you're hearing the noise floor that comes along with generating all that gain. The preamp's volume control can't save you here because it's UPSTREAM of all that gain. You wouldn’t hear so much "rush" noise if any of the following were true:

• More of your system’s gain was apportioned to BEFORE the volume control (but then you might risk overload conditions from "hot" sources)
• You insert attenuators between your preamp and amp (e.g. -10dB), raising preamp volume to compensate
• You use a preamp with either lower gain OR a higher signal-to-noise ratio. Many solid state preamps have a higher signal-to-noise for the same gain as tube amps, but there are some tube pres with very high ratios - like ARC Reference. My VAC Master is also pretty good here.
• Use an amp with less gain / lower sensitivity
• Use speakers with lower sensitivity /efficiency

I have the same issue with 96dB Tannoys (at 2.83V input = 1 Watt), tube preamps with "less than amazing" signal noise ratios and > 10dB gain, and power amps with mid-upper 20dBs of gain or more. Usually of the two - tube pre versus tube power - the tube preamp is by far the bigger culprit for noise, in my experience.

A lot of tube preamps and tube power amps have a lot of gain, because there are only so many circuits, and most of them use a front-end stage with lots of gain. 12AU7 and 6SN7 provide enough gain as is, but something like 12AT7 or 12AX7 gets crazy. The highest gain tube amps I’ve had started with 12AX7. And yes, microphony can be a big problem with many tube preamps too.