One of my favorite reharmonization tools in the arranging toolbox, when used sparingly, is the pivot reharm.
A pivot reharm is when you keep the original melodic note, but use it as a different part of an unrelated chord. For example, a B-flat can be the 5th of an E-flat major triad, but you can reharmonize it to be the third of a G-flat ma7 chord, or the sixth of a D-flat ma6 chord. I call it a pivot reharm because that’s what you’re doing – momentarily tonicizing a new tonal center bay pivoting around the melody note.
With the right voice leading, these can offer the harmonic complexity that you are seeking, by departing from diatonic chords while maintaining the familiar melody. It works best, indeed, when the chord is not diatonic. (Reharmonizing that E-flat Major triad as a B-flat major triad isn’t all that interesting.)
Gene Puerling was a master at this reharm tool, and used it to great effect particularly in his work with The Singers Unlimited. But the chord I think of as exemplifying this trick comes in Steve Zegree’s classic arrangement of the Gershwin song “Love Walked In.” In E-flat major, the melody of each A section begins on the third, G. When the melody returns halfway through the form, instead of E-flat major under the G, he harmonized it with a C(add2) chord. That major tonality from a fairly distant key consistently makes audiences sit up and listen intently, even if they are not harmonically engaged in the ins and outs of the arrangement.
Use it too often, and you’ll find you lose track of the tonal center, or accidentally modulate yourself fully to a new key. But used intentionally and the pivot reharm adds welcome harmonic interest to your reimagining of songs.
I am most apt to find pivot reharm moments in pieces as I noodle on the song. Honestly, they sometimes come when my undisciplined fingers lead me astray, but just as often they come from considering a key moment and how else I might harmonize a melody non-diatonically.