I can try a cap on the output. I intuitively had pondered that option too but then again the question "hmm, maybe a cap would help" crosses my mind very often 
Couldn't explain to myself why adding a capacitor might eliminate the coil whine in the power supply. The whine is due to a change in its internal switching frequency (or something in that vein), no?
I think I understand what you mean with your suggestion to add an always-on LED strip but that's not feasible for my situation. Entertaining workaround though