As you point out, once you've learned ear training, it's relatively straightforward to determine a melody, the chord, and even the bass notes (listen to the bass line, and treat it like its own melody). What can be harder, is the inner voices. It's not always easy to get these right, and, at least when I do it, it involves some amount of guesswork, and uncertainty.
The first thing to listen for is little countermelodies or riffs. Anything that's not the main melody, but has a distinct and discernible motion to it. Try and isolate these, sing them, analyze them (again, with ear training) to try and fill them out. Listen multiple times, possibly at a slower speed, if needed, and try to ignore the melody and bass (easier said than done). Sometimes it's easier to catch an inner part when it is moving from one pitch to another, and then you brain sort of "latches on" to it, and can hear it better.
You can use knowledge of the instrument to guide your intuition. For example, notes in a piano chord have to be so close together in order to be played (unless you start arpeggiating), and I believe the same is true with guitar chords. Many instruments have a more limited range as well, which can help to place the parts.
Knowledge of voice leading principles can also help to educate your guesses. For example, knowing that the third of a chord is rarely doubled, or that second inversion chords (with the fifth of the chord in the bass) are considered dissonant and therefore often avoided, except in cadential formulas or pedal points.
You also want to be aware of whether the parts sound dense (from a close voicing) or rather sparse (from an open voicing), and whether the parts are moving in similar motion, or contrary motion, or if some of them are just holding a note and sitting there (often done in string pads, for example).
In your example, you have a C in the melody, over an A minor chord. That means that, unless there are non-harmonic tones, everyone is playing an A, a C, or an E. You've already got the C in the melody, which is the third of the chord. In traditional voice leading (for vocal parts, and often in orchestral parts), the third of a chord is very often not doubled, so chances are good that there isn't another C out there, unless it's doubling the melody, or you have a very thick texture. Similarly, the bass is almost certainly playing an A. Somewhere in between, there is probably an E, and maybe another A.
One more thing you can do is come up with a hypothesis about what the voicing is, and then test it. For example, you might think that the E above middle C is the voicing being played at a certain point. Pause the song just before that point, and establish that E in your head. Play it on an instrument, sing it, whatever. Train your ears to focus on that frequency, then listen for it in the playback. Maybe even come up with a few possible voicings, and determine which one sounds the most like what your hearing.
And ultimately, if the voicing is so murky that you can't determine exactly what it is, then how important is it to know? If you pick a voicing that sounds "pretty close" (or it's so close that you can't tell if its right), just go with it. There are no voicing police to arrest you if you double the wrong note!