Spring is the best time to take cuttings. You could also layer some of your mother's plant. Take a long piece that is attached, scrape up some soil below, let the stem rest on the soil, place a small rock over it. Roots should develop.

Here is a link that might be useful: Gelsemium sempervirens

Oh, I'm sure layering would work---it works on almost anything. (*grin*) I was looking for specific methods folks have used to root actual cuttings. Sand? Soil? Perlite? Florists' foam? (I've had great success on a lot of things using florists' foam, but NOT Carolina jasmine.)

I root my cuttings in a clear gallon jug, set up like this:

In the bottom, I place 3" of really good potting soil.
Next I put in 3" of vermiculite.
Next, I soak the mediums and let set for a few days.
I am then ready to take cuttings and stick them.
I put the lid back on the jar for 24 hours, off 24 hours, etc.
Since the jar is clear, I can see when roots have formed.