Agarista populifolia = Leucothoe axillaris
Common names: Fetter-bush; Florida leucothoe; Coast leucothoe

I don't grow this species but two other species and propagation should be the same.
Semi hardwood cuttings taken most any time of the year, dipped into a rooting medium before sticking, should root readily.

Sorry, I don't remember what time of year I stuck cuttings but they rooted.

Hey Terramadre,
Thanks for the info.