Unfortunately, Louet does not offer a conversion kit to convert an old S10 spinning wheel to a double treadle system. This is because there have been many small changes over the years, making it impossible to make a standard conversion kit for it.
The S10 Concept, the latest version of the S10 (on the market since 2015) does have the option of changing the treadle system from single to double-treadle and vice versa.
If you're handy, or have access to a wood worker, you might be able to make the S10C's double-treadle base fit your old spinning wheel, but we can't give any guarantees on this kind of customization.
The base is fastened with a bolt in the upright of the wheel. With the S10 Concept, a special tapered groove has been made in the upright, which makes this connection very stable.
The footman and the footman to the wheel connections are otherwise the same.
Related article - What is the difference between double and single treadle wheels