Thank you Steve, that's highly valuable advice!
Latest version:

Redrew it at the larger original scale; tried to make the snake more stylized and graceful, and got rid of the ring.
Decided those before reading your latest edit: looks like we're on the same thought track!
I'll seriously consider the possibility of a headless snake too.
Greyscale (because of overlapping shapes) and pure black versions; sure gives a better idea of how shapes work together.
Also it might have to show acceptably on just any B&W desktop printer.
@ NaCl, Jeremy: yes it's Asclepius's rod - see my first post

Over here too (France) the rod appears in the most traditional forms; these look too rigid however in my (our) opinion. They're more suitable for really official documents, like parking permits...
My friend is not even insistent on the snake.
I think it's important.
In many of the
google refs which are specific caducei (

) for nurses, the rod is actually an
I for
Infirmier (nurse). We're not getting into that either.
For the fun, the one with a large
O is the midwives'! (and a lovely one too)