What makes a speaker sound great at low volume?

Lots of things. My speakers aren’t the most effecient and require a big amp. My system sounds very good at low volumes which is how I do most most of my listening. I’ve done some tweaking to lower the noise floor which is what I think has worked for me. Start with clean power, dedicated circuits, quality receptacles, quality cables, and cable placement. Nice rigid equipment stand, isolation devices between your components and shelves. I think it’s getting rid of the noise to start with. Speaker placement, your listening position and room treatment also play a big part. Guess I’m saying it’s not just looking at a pair of ultra efficient speakers because it can be done with not so effecient speakers as well. It takes time, experimenting and not being afraid of tweaks, such as fuses and etc.

My rec room speakers are a pair of vintage DCM TimeWindow 3's.  I set the volume on my McIntosh 31V A/V control center to an acceptable level and then use the 31V's EQ and McIntosh MC2600 power amp gain controls to tweak the sound.  This system is all analog.

In my music room, I have a NAD M12 and NAD M22 v2 pushing a pair of Tannoy dc8ti speakers.  This system is all digital, except for my Techniques SL 1600 MK II turntable.  I use the M12's volume control, which works OK - but , the treble and bass controls are no match for a good EQ like the 31V

I don't know if particular design attributes contribute to, or detract from, low level listening.  But, as a general observation, the best sounding systems at low volume tend to be high efficiency designs, particularly horn-based systems.  It may seem odd, but, those massive horn systems play better at low volume than at high volume (this might explain why so many Japanese audiophiles are attracted to horns even if they have neighbors that are very close by).

In part, high efficiency allows for the use of low-powered amps.  Low-powered amps, particularly tube amps, sound very lively when played at modest levels, while high-powered stuff tends to sound dead at low volume.

chayro: Love your first sentence ...

OP: I don't know if you're considering vintage speakers as well, to solve your low-level, listening issue, but I have a pair of Kef R107's, and they produce a wonderful sound-stage no matter where you are in the room. They are supplied with a small EQ adjuster that is electronically, part of the crossover. At low volumes I just boost the Contour adjustment dial. (This is the only time I touch the contour dial). This seems to add what's required to cover all the necessary frequencies, and the music sounds great.
Most people seem to like the 107s, but some just don't. They're a little different.
Anyway, just my two cents ...