I would say, to best understand harmonization of all music, it would help you to move beyond this concept:
When choosing chords in a tonal, diatonic concept, the formula is somewhat clear to me. If it's a note in the scale, choose I, IV or V if in doubt.
I suspect that most songwriters and composers are not at all thinking like that. It's not like they have a melody and they are just searching for chords that work with it. It's that the melody and chords work together to say what they want to say, and only one chord is the right chord for that statement combined with the melody note.
For me, it was Leonard Bernstein who helped me understand this by drawing an analogy between the notes of the melody being like nouns and the chords/harmony being like adjectives.
Suppose a melody note is like the noun "apple". Then you can talk about different kinds of apples by changing the harmony supporting the note. Like a red apple, or a rotten apple, or a poisoned apple, or a Cox's Orange Pippin just like your mom used to bring home from the orchard in early fall when you were living in upstate New York at the age of seven.
So even in diatonic composition, we would have a very small musical vocabulary and we wouldn't be expressing ourselves well if we A) had "doubt" about what chord to use and B) pretty much kept to I, IV or V because of that doubt.
Obviously there are many effective, evocative songs that only use I, IV, and V, or even simpler harmonizations. Two things: Sometimes the musical equivalent of baby talk is exactly what is intended, either to highlight the universal and ageless nature of the message or to leave the listener able to focus on lyrics or something like that. Secondly, even when the harmony for a piece is very simple, often the melody that goes with that harmony is all the more complex. I think delta and Chicago blues are excellent examples of very straightforward harmonies backing subtle and not quite diatonic melodies.
So that suggests approaching your question from the opposite direction. What if we asked, "How can one make a chromatic melody work over a I - IV - V chord progression?" Well, that's kinda like what blues is!
If you have a keyboard instrument or some ability to do so, I highly suggest playing or programming a single chord to repeat or sustain, and then slowly play a chromatic scale above that chord, listening for how each note of the scale interacts with the chord. Don't listen for whether the notes "work" or if they sound "right" or "wrong" with the chord, listen to how everything sounds together and what kind of feel you get from it. It can be very interesting to continue the scale an octave higher than you started it, to see how the distance between the "melody" notes and the chord makes a difference.
So really, any chord can be used under any melody note, as long as it supports the musical message and fits into the overall context.
Beyond that, chromatic music can be written either semi- or fully contrapuntally, where there isn't as much of a harmony/melody distinction, but intertwining passages that form both a melodic and harmonic picture. And there things like chord planing/parallel chords (as mentioned in the comments) where the chords are kind of playing the melody.
To summarize, I think it would help you immensely to try to reset or transcend the notion of "writing a melody and then finding chords that fit with it". Perhaps a good first evolution of that idea is "writing a melody and then thinking about each note of that melody and what 'adjectives' best modify that note".
Another way to break out of the melody/chord concept of composition is to write a melody, then write a bass line that you like along with the melody, and then add a third line in between them. And/or mix up the order in which you write them. And definitely think about each of the three lines of music standing on its own (e.g., don't let yourself write a bass line that is just the same note for three measures and then a different note for the fourth).