Ways to Improve a Recording
Something you can try is importing the song into audio editor software like Audacity and modifying the recording to highlight the piano. If it's a stereo recording, you can isolate the channel in which piano is more prevalent and increase that channel volume to make the piano louder relative to the band. Another thing you can try is to adjust the equalizer settings to highlight the piano frequencies. GarageBand has equalizer presets and adjustment tools specifically for things like this. This could entail listening to the recording through GarageBand, watching the graph for likely piano frequencies, and increasing those piano frequencies in order to highlight the chord voicing. Finding the right equalizer settings likely will involve some guesswork, and it obviously can change based on things like recording quality, range of voicing, and piano being used. I haven't done this enough to offer guiding principles for piano transcriptions specifically, but perhaps others will chime in.
Part of this equalizer work can be done in tandem with frequency graphs from other parts of the song. If you listen to a clip of a drum solo and identify the higher frequencies that are simply produced by the cymbals, then you can reduce the levels on those frequencies to emphasize just the piano.
If All Else Fails
Sometimes there is no way to improve the quality of the recording, and you're simply stuck with a scenario where the certain notes cannot be obtained from the piano. Here are the tricks I usually try (some of which you've already described) to fill in the gaps when dealing with a low-quality recording. First, I look to see if the "holes" (notes that were played but were inaudible) fit within any broader patterns. This can apply to transcribing a melody/solo line--for example, I'll look for arpeggios, scale runs, and scale patterns that seem to be missing a note that was likely played but simply cannot be heard on the low-quality recording. This technique can also extend to transcribing chords--sometimes there are obvious-enough gaps in voice leading, inner voice movement, upper structure triads, etc. to infer that certain notes that were likely played but not audible.
These are some of the ways to follow logic, as you nicely put it. But if this fails, then in my experience, the best next thing is to become extremely familiar with that pianist's playing. My first big transcription project was to transcribe all of Kenny Barron's album Green Chimneys. Halfway through the project, I could somewhat regularly predict the end of a solo line/lick after hearing only the beginning of the line/lick. Intuition was kicking in--and this intuition was owed to the other transcriptions I had done of other solos earlier on the same album. (I was only transcribing the solo lines.) But I've found something similar to be true with Chick Corea voicings and Bill Evans voicings (though to a lesser extent because I've spent less time transcribing their chord voicings).
Unfortunately, most people don't have the luxury of transcribing 5 other songs purely for the purpose of aiding an original transcription with inaudible/indistinct notes. So, a more direct route is to find other versions of the same song performed by that same pianist. As you'll appreciate, some pianists are fairly fixed in their solo arrangements, and from one recording to the next, there won't be much deviation in the notes being played. Listening to other live versions or studio versions of the same solo piano piece can offer insight into the hard-to-hear (or impossible-to-hear) notes that have been lost to a low-quality recording.
This is an effective tool for voicings. My experience has been that voicings are re-used even more than solo lines. If you know of a second recording of the same song by the same pianist, you can start by limiting your guesswork to voicings that appear in the higher-quality recording. In many instances, there will be a match. This type of matching greatly reduces the number of possible voicings you're considering for a given chord. For example, if you know that an
A7sus chord is being played, but you can't determine the specific notes, you can start by looking at the ways the pianist voices
A7sus on the other recording, and check for a likely match. Even if you're not doing this for every chord, this can still be a tedious process, but over time the voicings become internalized, the same way that one can internalize improvisational lines.
Using Poinciana as an example, you might try transcribing one or more of these recordings (example 1, example 2, example 3, example 4) where Ahmad Jamal is playing the same song but at a different concert. At first glance, it sounds like his chord voicings over the melody are mostly the same, even in the more recent concert that occurred decase after the '58 recording. There won't be this same 1-to-1 match in the voicings he uses when soloing, but I bet there is still a lot of overlap.
This may not apply to you so much as other readers, but it will be equally useful to focus on continued ear-training. The process described above can work in moderation, when dealing with a few chords here or there that simply are not discernible from the raw recording. But trying to engage in this process for every chord in a recording is probably not manageable. If that's the scenario one finds oneself in, then either the recording is un-salvageable or more ear-training work is needed. When transcribing chords, it helps to be able to identify all of the notes in a familiar chord when played clearly. If one isn't able to recognize all of the notes in a rootless
C7#9b13 voicing, then it's going to be even more difficult to do this with a less-than-perfect recording. Perfecting that skill (through ongoing ear-training) makes a big difference when tackling poor-quality recordings.