I know that I can iterate through the cookies in a cookiejar, and this would allow me to find a cookie with a particular name - but does the CookieJar object itself have any
You can also use dict_from_cookiejar, which returns a key/value dictionary from a CookieJar. Something like:
my_cookies = requests.utils.dict_from_cookiejar(s.cookies)
and then access your cookie value by key.