Apartment Drama

Headphones usually have capabilities to provide better definition and imaging and the best piecefull solution.
My other solution would be to advice neighbors to appear next time knocking at your door with presence of police officer(s) only. Cops less-likely will open the case or even will care to arrive for minor neighbor troubles thus leaving your neighbor 'grouchy' and somewhat 'upset' and probably seeking some mental help. Had similar neighbor by myself always knocking at my door and complaining about my noisy kids playing in the middle of the day.
While you being PIA to your neighbors you can always place your speakers on cardboard boxes as Elizabeth suggested. I found Egg boxes work best and somewhat as good as dedicated speaker stands.

What was the decibel level you were listening at?

Talk to your neighbor. Find out if there's a real problem with the sound or if it's something else. A few people are totally unreasonable nearly all the time, but most people are kinda' nice if you're nice to them.

BTW, when I rented I would always tell the landlord that I play music often and loudly. I don't think I ever lost out on an apartment because of it, but the landlords were thankful and sometimes made special arrangements for me.

I lived in apts my whole life and hated those neighbors with the thumping coming thru the building. Now that I have a house I can be lord of the manor.
Elizabeth has it right, it's the bass, you'll never be able to use cabinet spkrs in that apt. Maybe some small monitors on strands like totems. But she has a proven solution...why not give some Maggies a try. And a good set of headphones for late night listening.

But you will never be able to crank it up again in this current apt.

I disagree with Elizabeth only to a certain extent. Yes, we all have a right to
privacy in our apt.; however, everyone that chooses to (or has to) live in an
apt. has an obligation to remember that, because it is an apt, one cannot
expect the kind of absolute privacy that one gets in a private house. Both
sides need to be reasonable and compromise. That is not to say that you
will persevere in this case, but I would start by presenting yourself as
reasonable and fair to your neighbor: tell him that you are concerned and
want to find a solution. Ask him to allow you to go to his apt in order to
hear what it is he is hearing (while your stereo plays from above at a
normal level). Be reasonable and put yourself in his shoes; maybe the
problem is real. If he is being totally unreasonable because what he is
hearing is truly subtle and very distant sounding, then talk to him and try to
reason with him. If he is completely unreceptive then personally I would
escalate things to reliance on what the local ordinances mandate; but,
always take the high road and avoid nastiness as much as possible. If
there really is a problem (put your own bias aside as much as possible),
then tell him you are working on finding a solution and would appreciate
some patience while you do so. Then try some of the remedies that will be
recommended and possibly ask the neighbor about your respective
schedules; there may be times of the week when he is not around and you
are and you can rock out. Invite him over to listen to your stereo; keep
things friendly as much as possible. Good luck.

If this person has non-traditional sleeping hours, that may account for why 3 pm was a problem. You really do need to investigate, as described above, to find out how much of a problem your system is really causing.

If the problem apartment is the one below, you might reduce some of the transmitted sound by using an isolation platform under the speaker to absorb vibrations (don't use cones that couple the speaker/stand to the floor.

Dipole speaker so a much better job of focusing the soundfield to a listening area between the speakers. This is because the opposite front and rear polarity causes cancellation at the sides. The large panel types also act like a line source which means that sound levels do not fall off as much for any given distance between the listener and the speaker; that means you can play with lower overall sound energy in the room (again because the sound is concentrated at the listener). I went from dipole speakers to regular dynamic speakers and the difference in sound that leaks thoughout the house was startling. You could look at something like Magnepan 1.7s which are line-type dipoles and don't produce too much bass.

Small stand-mounted speakers will also reduce the amount of bass energy and that might also do the trick. I would look at something like Proac Tablettes.

There is not much one can do in the way of room treatments that would really work. If your place is not carpeted, a thick carpet, particularly with a good pad underneath, will cut down on floor transmission of sound. Make sure all doors are closed and tightly sealed when you play your system to minimize the amount of sound that goes into other rooms of your place.

Good luck.