[morzh]

The filter caps didn't do it.

Then I put the cap between the earlier mentioned transformer #22's pins that go to the 26 tube grid cap and #24 inductance. I thought of trying to eliminate the common mode noise that could somehow get through via xfmr's parasitic capacitance by shorting it via that cap.
This did have an effect: the hum went down significantly, and the distortion I complained before all but disappeared.
I am not sure I understand where the noise comes through in first place: it is not from detector, as it remains if the detector tube is out or its grid is shorted, but the xfmr is shielded. Then touching the chassis anywhere including the xfmr shield would increase the hum.
This all could be due to the different cathode potentials, but they are all tied together by that big resistor and decoupled with capacitors.
If it is what I think (CM noise) the same cap from the other primary pin and the GND should've worked too, but it didn't.

I even thought of eliminating the transformer altogether as it can be simply replaced with a cap and two resistors, but decided to not do it.

So....I decided to take it and be done with it.
Tomorrow will put it into the cabinet again and see how it works.

It sure sounds much nicer now.