Summertime is in A minor and has a 16-bar structure divided into two 8-bar sections (A and A’). With this in mind, here are some ideas for reharmonization:

1. Explore modal interchange: Since the tune moves briefly to C major (the relative major), you could introduce more chords from C major throughout.
2. Add secondary dominants: The original has one secondary dominant (A7alt leading to Dm7). You could introduce more secondary dominants to create tension and resolution.
3. Tritone substitutions: Replace some dominant chords with their tritone substitutes for a more modern sound.
4. Extend harmonies: Where appropriate, you could extend some chords to 9ths, 11ths, or 13ths for added color.
5. Use diminished passing chords: These can be inserted between existing chords to create more movement.
6. Explore minor line clichés: Since the tune is primarily in A minor, you could use descending or ascending chromatic lines in the bass to connect chords.
7. Alter the harmonic rhythm: You could change how long each chord is held, creating more or less movement in certain sections.

Remember, the goal of reharmonization is to enhance the original while maintaining its essence. It’s important to preserve key structural points and the overall feel of the piece.