If you are looking for a French name with a similar meaning or sound, you could consider:
* Jéhanne: A French variant of "Johanna," meaning "God is gracious."
* Jacqueline: A French name meaning "supplanter" or "God is merciful."
* Jocelyne: A French name meaning "cheerful" or "God is my oath."
* Jessamine: A French name meaning "jasmine flower."
Ultimately, the best French equivalent for "Jessamyn" depends on the specific meaning and sound you are looking for.