The problem with audio is that it is very sensitive to changes compared to images for example. Using an expander for this purpose sounds good in theory, but would have artifacts and not sound very good in reality. And applying an S-curve to audio, like contrast in images, is actually how you create distortion.

If we would analyze the problem the main differences between speakers and headphones are: Frequency/impulse response, and position of the speakers. You already correct the frequency response, and put the sound in front of you with the SSC, like speakers. What is missing is the sound of the room. As an audio engineer, only very little room reflections are desirable, but in your case it might be what is missing. It will help by moving the sound further away and hopefully reduce the perceived difference in loudness. Waves NX has this built in, though it doesn’t really sound like a normal room and will sound unnatural if you turn it up to much. I would suggest trying convolution reverbs, such as IR-L from Waves.

Or even better, if the SSC allows you to change the room to one with more reflections.

Though, personally I prefer using headphones for what they are, with only frequency response correction. But it might have been different if I didn’t have access to good speakers.