Sorry for the very-late reply, but I haven't visited here in some time.
I tried to find a diagram or specification on the transmitter cord myself, but was never able to. Archos appears to have "burned" all documentation... or at least has it buried in a very, very deep hole.
Your best bet would be to find someone who has such a cable and is willing to part with it, so that you can perform an "autopsy" on it. I have a cable, but I need it, as I still use my 500GB unit during travel in my car as my "infinite audio selection" player. (If someone were to produce a modern tablet with 500GB or better internal storage, in a similar form factor, the Archos would be retired immediately, however, no matter how much I prefer Archos' audio player front-end!)
(I own a Galaxy S7, with a 200GB card, too... but that's not remotely close to 500GB, is it?)
ANYWAY... if you can find one, the "autopsy" would be relatively trivial... disassemble it (which will, unfortunately, be a "destructive" process, so do it CAREFULLY!) and trace each line to where it leads to.
As I understand, activating the FM transmitter function (which is entirely contained within the media player unit itself, except for the antenna!) requires a resistor of a specific rating to bridge two of the contacts in the connector. So, that, and two contacts which go to lines in the power cable which serve as antenna elements... is all you really need to know.
I suspect that multiple pins (likely 4 pins, 2x each for each of the following) in the connector go to the 12V+ and G (or is it 12V- and G?) points in the cigarette-lighter plug, and the rest of the connections are unconnected. (Except for the resistor and the antenna lines, I mean.)
If I had a spare I didn't mind parting with, I could diagnose this and come up with a wiring diagram which would allow anyone to make one.
If you're handy, and have the time and inclination, YOU could do this, AND even produce some for sale (on Ebay?) for anyone else who wants them? It would be pretty easy to do.