Put a small (1- to 2-qt.) pot on the stovetop. Pour in wine and add cinnamon stick to wine. Adding remaining ingredients to a large stainless steel mesh tea-infuser ball (or make a bag with a fine-weave cheesecloth and twine), working quickly to close the infuser ball once you add the jam (as it will begin dripping). Add infuser ball to the pot of warming wine. Turn heat to medium-low.
Ladle wine over infuser ball, then cover wine and heat for 5 minutes. After 5 minutes, ladle wine over infuser ball again and heat, covered, for 5 more minutes. After a total of 10 minutes, remove cinnamon stick, then repeat ladling of wine over infuser ball. Taste the wine at this point, and add some superfine baking sugar if you like a sweeter mulled wine. Continue to stir and heat for 5 to 10 minutes (for a maximum of 20 minutes total heating time).
*Look for a loose (with little pectin) blackberry jam with some sugar; this type of jam will melt into your wine when heated, while sugar-free fruit juice-flavored jams tend to get gummy instead of dissolving into the wine.