This could simply be a tiny, insignificant detail.
Had you noticed that he had two caps wtih him, according to the postmortem report?
The peaked Barbour cap is on the ground and bloodstained?
A flat cap is in the game pocket. NCH/17/5 in the postmortem report.
Of no meaning? He could simply have forgotten about the cap in the game pocket?
Or was the Barbour cap the obvious thing for a third party to put on display, unaware of Dr. Kelly's flat cap in the game pocket?