[...] I guess you meant lowering general internal noises from the DAC itself, not the digital volume alone. [...]
I'm only thinking that between the DAC and the headphones jack there are lot of active and passive components that generate electrical and thermal noise. Placing a resistive potentiometer or ladder array between DAC's output stage and gain stage of the headphones amplifier will reduce the background noise. Of course, this may only help with >100dB/mW low-impedance headphones.
My test IEM's are Superlux HD381f having 16 Ohms internal DC-resistance, 103dB/mw, 120.5dB/V sensitivity. To drive them with peaks around 100dB (normal peaks for an average listening of 80-85dBFS) the amplifier needs to apply
90mV RMS, so an amplifier with
less than 9uV of background noise will be needed for an attenuation of 80dB (usually we don't really hear a noise 80dB lower than music's peaks). So it's really hard for an amplifier to provide such low numbers, especially if internal gain is high (>5X) and there's no resistive divider in there to lower the output volume.
P.S.: With volume to the max. (input RCA shorted, no music playing) Burson PLAY v1.6 with AD8599/AD797 and custom linear PSU is having the lowest background noise from all my amps (compared with Matrix M-Stage HPA-3B with medium-gain, Objective2 with a gain of 3.3X, Burson FUN, ASUS Essence One). For the above "test", the analogue volume control was set to around 7V RMS with 1KHz sines, then all input sources went muted, so I can wear the sensitive IEMs and A/B test the internal noise. If notching down the analogue volume controls down to zero value, then Objective2, HPA-3B, Essence One and FUN are the winners, in that specific order, then follows the PLAY combo. Of course, the noise difference I'm speaking about are so tiny that during the day this test can't take place, so only during a very quiet night this can happen.