Maybe a picture paints a thousand words...
Attached is a graph showing my main outdoor temp and the solar jar temp for the last 24 hours.
For most of the night (sunset ~19:45, sunrise 6:53) the solar was around 0.2 to 0.3 above outdoor. I'm assuming that that represents 'overcast' for my set up (the small positive difference may just be a matter of calibration differences between the 2 sensors).
A few times the solar dropped below the outdoor, e.g. just before 23:00 and between 00:30 and 02:00. I'm assuming that those times represent a clearer (but not totally clear) sky.
It so happens that I did look out just after 11pm and the moon was clearly visible, but with halo, for a short while - so maybe broken cloud? The general drop in temp at those times would also support the idea of a break in the cloud.
Also attached a picture of the "modified" solar jar.
I got a piece of black plastic, wrapped that around the bottom half of the jar, then wrapped black tape around that - the idea of the inside plastic layer being that if I decide to remove the black bottom I won't be left with a mess of adhesive to clear off the jar.

We've not had any 'proper' sunshine since I made the mod, but so far as I can tell the black bottom seems to have had no adverse affect daytime use.