I love this trick for reharmonising a melody note, both for playing with harmony in jazz standards and writing simple songs...

Further Reading...

So lets just recap on the theory for each chord... but remember that I do think it's better to just use your eyes and ears for this!!

Our melody note is E... there are many more chords available than those listed, these are just common ones to try out.

C Bass note
The note E is the Major 3rd, so any type of major chord will fit nice, C, Cmaj7, Cmaj6, Cmaj69, C6, etc.

C# Bass Note
Now the note E is the minor 3rd, so minor type chords fit, Cmin, Cmin7, Cmin9, Cmin6, etc.

D Bass Note
The note E is the 9th, so both major minor and dominant chords works with the 9th so D9, Dmaj9, Dmin9, etc.

D# Bass Note
So this one is a bit awkward cos the melody note is now the b9, which will most commonly be used in an altered dominant, do D#7b9.

E Bass Note
Well the bass note and melody are the same, so ANY E chord will work!

F Bass Note
The E is now functioning as the Major 7th, so chords would be Fmaj7, FminMaj7, Fmin6Maj7.

F# Bass Note
In this harmony the E is the dominant 7th so will work over F# domininat or Min7, of F#7 and F#min7

G Bass Note
E is the 6th now so G13, GMaj6 or Gmin6 will be the best options...

G# Bass Note
We're getting altered again, and the E is the Sharp 5 which is most common in altered chords like G#7#5.

A Bass Note
The E is now the 5th so there are many chords that will work, Amaj, Amin, A7, Amin7, Amaj7 and lots more...

Bb Bass Note
E is the super cool #11 now, love this sound, most common in C/Bb or BbMaj7#11.

B Bass Note
The harmony here is a little more limited with the E being the 4th, so B11 or Bsus4.

There you go - head out and explore this idea now on your own!! :) Good luck.