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Thanks, Rainer, for your spot-on observations! According to the manufacturer, my headphones have an impedance of 25 Ohms and a sensitivity of 121dB/mW. Yesterday, I experimented a bit with attenuators and managed to achieve satisfactory results by connecting the headphones through an attenuator of about -20dB.
Previously, I was using the X6100 with headphones at a volume level of 1–7 (in R1CBU), but now I can comfortably use it at a volume level of 1–30, with a significant improvement in S/N!

Now I'm wondering how much the S/N could be improved on the line between the audio codec and the internal sound card if proper level and impedance matching were applied there.

So I can say that simply using an attenuator in my setup brought very tangible results!