That "chime" (ting-pshhhhh) is extremely typical of 6H30, and is quite distinct to that tube type. If it makes you feel better, other brands of 6H30 preamps have the exact same deal - any type of click relay inside such a unit will set it off. The biggest variant is how much gain. Rogue’s Hera II had much higher gain than Ref 6 and was almost unbearable in my system (high gain amps and high sensitivity speakers made for a bad combo here) - in fact it got to the point I could hear the damn chime excited between notes sometimes, at loud playback volumes.
The Ref 6 is below the threshold where you have to worry about this getting excited during normal playback. The relays just set it off because they’re right there inside the box, on the same springy PCB as the tubes. You’re not activating those relays during music playback, so no worries.
As with all tubes, the microphonics can get worse with age, and on/off heat cycles. All the critical tube elements are pressed through those thin mica wafers, and as those holes get stretched you get the inevitable result. Scrupulous sellers like ARC and Upscale will also do a great job screening these tubes and getting you the best, quietest new set.
Tube dampers don’t do a goddamn thing here (I tried Herbie’s etc), so don’t waste any money there. They don’t have enough mass to be effective, nor are they attached directly to the internal elements which do the ringing. I also did not find exotic external isolation platforms or component dampers to have any significant effect.
If you’re concerned that it seems like a sudden onset in your case, it was probably there before and you’re more sensitive to it now. Sometimes the relay will randomly have a louder click / pop effect, and that’s it.
If it still bothers you, the only solution is to move to a preamp that uses a different tube type. The plates of 6H30 are very large for the gain rendered, and physically this is what happens in these scenarios. Tubes like 6922, 12A*7 have much smaller plate structures and can be dead quiet even next to relays (when tightly selected). 6SN7 are another large plate, medium-mu tube that can be very problematic in preamps, but their "chime" sounds much different to 6H30 (more of a tink-thuump vaguely like tapping a microphone). Fewer of the 6SN7 based preamps seem to have relay based switching (old world mechanics for an old world tube type), so I guess fewer people complain there ;)