Howzabout rather than worry about what "key", listen to hear what "tonal center", in other words, what chord sounds most like home base? Most of us, I think, agree that A sounds like the resting place in the progression.
You can then tweak your improv line to suit whatever the chord are (or not, if you like that sound). G's on everything but the E which would get a G#. Or stick to G if you like that and get a bluesy #9 thing. Or play whatever you hear and don't worry...
I play a G chord in bar 4 of the A part of Old Joe Clark because I like the vibe. The melody note is an E, it adds up to a G6 sound, it may not be "no part o' nothin" but I like it, and for my taste the E chord is too vanilla- I like the 'modal sound' of the G chord.
My changes for OJC:
A/A/A/G/A/A/AG/A
A/A/A/GD/A /A/AG/A
This could be called reharmonization, or personal preference, or stubbornness
The ending phrase is usually C# B A C# B A G# B A- I play G rather than G# there to fit the chord. One little note can have a powerful affect, don't it?
If I were jamming with a bunch of folks who play E, I'd play it too, by if I were the band leader, I'd call the chords before we played it.
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