What makes a speaker sound great at low volume?

Yes - look for a speaker that sounds good at low volumes. I’m not being sarcastic. One of the many mistakes I found that people make is doing demos at loud volumes and then deadening their ears. I think ears react to sound like pupils to light. If you go into a dark room - the eyes open up and then you can see. If you start listening at a volume that’s a bit too soft, your ears open up and the volume gets more normal. Then if you kick up the volume just a smidge, it sounds dynamic and full. If you listen at low volumes you also might consider an amplifier with tone controls to allow you to add a bit of bottom and top to the mix when you’re listening softly. I’ve worked pretty hard at putting together a low-volume system and there’s no real secret, IMO. Demo the speaker in your home and see if it sounds good at low volumes. That’s pretty much it.
In fact - try the low-volume experiment with your current system.  Start a little too soft, listen for a few minutes until it feels comfortable and then turn it up just 1 notch.  Maybe you're ok as is and it's just your listening technique.  That saves a few bucks.  Good luck. 

One thing I'd say you likely Don't need (necessarily) are expensive, overdesigned drivers. Even ordinary paper cones are quite competent and capable of very nice combinations of musicality and detail/resolution. An exception in your case might be planar/electrostatic/ribbon designs. These, if they fit your budget, can offer even more separation/clarity/transparency and so forth.

Where I think the most design money must be spent in your case might be on the crossovers and it will be a challenge for you perhaps to find an example with nice, ordinary drivers combined with excellent (expensive) crossovers at an affordable, or entry-level price. Of all the design components in a speaker, I'd say generally that the crossover is the single biggest obstacle to good sound. But this is only because most manufacturers end up giving it short shrift - they likely could do better, but they simply don't. Most people don't realize just how important crossovers are to the kind of sound they are looking for and most makers seem content to let that sleeping dog lie, if you will, rather than try to educate the buyer. But for you, crossoverless designs are at least something to consider and to go and hear, if you have the opportunity. Single-driver designs stand to save you some big bucks and a long time looking by doing away with the crossovers altogether. The only real catch to look for might be the relatively rolled-off highs since no single driver is truly considered full-range.Crossovers may take the single biggest hit to sound quality in the overall resolution/clarity/transparency/coherency package - perhaps especially the coherency - whether it be dynamic, tonal, vocal intelligibility or spatial coherency.

All that I'd add is that while you'd want speaker sensitivity to be modestly high, you certainly don't want it too, too high, or you might be dealing with a system hum that is most annoying at low volume levels and that you then can't get rid of without swapping out gear. In the low 90's might be ok, but 100+, almost certainly not.

In terms of what metrics to look for, in my experience the speakers with very high sensitivity, >96dB, combined with high load impedance, >10ohms, tend to sound more detailed and full at lower listening volumes. I have a pair of single-driver full range horn loaded speakers in my upstairs converted listening room paired with a 10wpc integrated amplifier and its perfect for (almost) near field low volume late night listening.

Crossovers are the most ignored, but most important consideration, IMHO. I recently went through the same process you’re going through now. I ended up with the 2 way Joseph Audio Pulsars, which provide everything I need ... great crossover (a Modafferi based and JA modified infinite slope), fabulous sound at low and high volumes, low distortion, and great looks. Couple them with a good sub and you’re good to go. BTW, I went from full range B&Ws to the 2 way Pulsars, and can say that I much prefer the more coherent sound of the 2 way with a sub than the full range speakers. 2 way speakers can "disappear" in a way that full range can’t. Of course, your room size will have a major impact on which way to go as far as speaker size is concerned.