No, the volume is determined by the choice of electronic components and that requires a power supply within certain tolerances.
If the Babyface hadn't been designed to run on USB2 bus power alone, the circuits could have been designed differently so that it would have been louder in the end. But mobility is one of the main use cases for this device.
The external power supply is useful in such cases when the quality of the power supply on the PC or laptop is not ok, this can bring more stability in operation.
So you say you have the original Babyface. Which headphones do you have and to which output on the Babyface do you connect them?
You know that the new Babyface Pro and Babyface Pro FS has improvements here? The manual tells:
"The output channels 3/4 feed two Phones outputs via two independent driver circuits, optimized for high (TRS 1/4”, 6.35 mm) and low impedance (TRS 1/8”, 3.5 mm) headphones."
The volume is 60/90 mW. Maybe time to think about an update and try. Alternatively, if your budget allows, you could also think about connecting an ADI-2 FS or ADI-2 DAC FS via optical SPDIF or another headphone amp of your choice, analog or digital.
BR Ramses - UFX III, 12Mic, XTC, ADI-2 Pro FS R BE, RayDAT, X10SRi-F, E5-1680v4, Win10Pro22H2, Cub14