What makes a speaker perform well at low volume?

Second the good amp but add a very neutral cable

IME, the amp, the speaker and the cable all make a difference. Tube amps can sound excellent at low volumes, but I've heard a couple of SS amps that can too. Some speakers are better than others at low volume, more efficient speaker generally being better. My Tannoys were outstanding at low volumes maintaining life and presence, my AN E's less so which seem to lose life at very low volumes (and are actually efficient speakers at 94 db). But this last point I think could be related to the cable. I'm running 17 feet of speaker cable and something tells me that the long length of cable is not good for low volumes. I have not been able to try the E's on shorter cable.

Another good low volume pair was my JMR Trentes with the Resolution Audio s30 integrated amp. This amp is based on a DNM design which is noted for maintaining life at low volumes. Conversely, I had a Blue Circle amp that went dead at low volumes.

Three words: Low level detail. Sometimes the problem lies in the passive parts of your louspeakers (binding posts, wiring, crossovers, etc). sometimes it's the drivers themselves. Sometimes the problem is farther up the signal chain. And yes, high-efficiency/high-sensitivity speakers, especially single driver speakers, and well executed multidriver speakers are usually good in this regard. It also helps to have speakers that sound coherent (as cut from the same sonic cloth) over as much of the volume range as possible.

I don't have all of the answers. I've only been at this for less than 4 years.

Lack of hysteresis provides good performance at low volume. A speaker has to settle exactly at the same resting point every time there is "0" signal. Most do not and they "oilcan" to some extent. This muddies low volume inputs.

I was not able to "hear" everything at low volume until I replaced my pre-amp with a Tom Evans Vibe and Pulse. This suggests to me that every component may have something to do with this. Any weakness can possibly hamper the ability to play properly at a low volume.