I think it's really helpful to try and internalise the sound of various common jazz chord qualities. It is a lot easier to hear the complicated harmonies when you have a large mental database of sounds from which to work. For example, obviously any jazz musician should know by heart all the seventh chords, but some great useful sounds to know are:
- 7♯11 (7♭5)
- maj7♯11 (maj7♭5)
- +9 (9♯5)
If you are familiar with these sounds, a lot of jazz harmonies will seemingly transcribe themselves, and the more complicated chords can mostly be related back to these chords above. Now, dominant chords can be altered nearly beyond recognition, and you'll occasionally hear some chords with two or three crazy alterations, but with a couple listens it should be much easier to pick out individual notes. If you're not sure, the bass note and the top note are often the easiest to hear.
The reason why it's important to know those basic harmonies above is because sometimes on the recording it won't be possible to hear every note and piece together the chord. If you can just listen and hear "okay, that's a minor 9 chord, to a diminished 7th chord, and oh, 13♭9, aaaand back to maj7", you can figure out the chord roots easily by ear. With practice, this can allow one to really play some complex stuff by ear and transcribe jazz music much more efficiently. Of course, it'll take time and effort to develop one's ear to that point, but with regular active listening to jazz music, it shouldn't be too hard to do.
Jazz is also a very functional genre; chords tend to follow predictable patterns. By far the most important pattern in the genre is the ii V I and its minor variant ii° V i. When you listen to a lot of jazz music, you start to hear the tonic, predominant, and dominant roles on the harmonies, which will help determine both the quality of the chord and its root. As a simple example, suppose you know a chord progression is Cmaj7, E♭°7, X, and G7, but you can't tell what's being played at X. The only real option for that chord X is some kind of Dm7 chord, unless the E♭°7 chord is functioning differently than the way I labelled it (actually, maybe a D7 of some sort could work). Knowing chord progressions will let you "know your options" at a moment in the piece, so to speak. Secondary Dominants and Tritone Substitutions also are really important concepts to understand, as they can easily explain some of the more challenging harmonies to hear. This is why it's very important to be up on your theory to transcribe jazz music: if you have to get out your circle of fifths to determine the V of every E♭ chord, you're much too slow. Also, jazz gets written in plenty of keys, and uses chords on every root, so better get away from just "thinking in C" every time.