The best way I've found to "Jazz up" a simple tune is to write the song
backwards.
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.
