Jazzing up a simple tune
In other words I decide where I want to end up and then use the circle of 4ths (with tritone substitutions) to go back as far as I can while still (mostly) preserving the melody.
You'll see that the following harmonization (in G) of the last part of "Amazing Grace" uses the principles described above.
The melody notes are in parentheses followed by the chord to harmonize with.
once (E) was (D) G13
lo- (B) -ost (G) CMaj69
but (G) now (G) I'm (G) F9
fou- (E) -ound (D) was (E) E13
blind (G) but (G) Am9
now (Bb) I (A) Ab9
see (G) G6/9
The tritone substitutions in this circular progression are:
chord 4 (the E13), and
chord 6 (the Ab9)
You'll notice that I "bluesed" the first melody note over the Ab9 (Bb) which then moves to an A and turns the harmony into an Ab7b9.
It took me more time to write this down than what it did to figure out this great sounding cool progression. My decision making process was ruled by which chord (the straight circular chord or its tritone substitute) went better with the melody line.
The next step now is to throw the melody out the window and find a dozen solo ideas over this progression.