Put the plums, 2 cups of sugar, lemon juice and rum into a large saucepan then bring to a boil while mashing with a potato masher. Reduce heat and allow to simmer, stirring often, about 30 minutes. With a stick blender, carefully blend in the saucepan until all skins are smooth.
Stir in the ginger, taste test then add additional sugar, if needed. Sprinkle in the a small amount of cinnamon. Continue to simmer until reduced and thick, about 45 minutes to an hour. Allow to cool, then bottle or freeze.
I used some yellow plums that a friend gave me (that I have never seen before, but tasted the same as the purple ones), so your jam may be darker than mine, but will taste just as good.