Preheat the oven to 350°F. Butter and flour or line a muffin pan with paper liners; set aside.
Add flour, salt, baking powder, and cinnamon to a medium-sized bowl. Whisk to combine, then set aside.
Using a mixer, cream butter on medium until fluffy for about one minute. Add sugar and beat for another minute. Crack eggs, one at a time and add to the bowl, mix after each addition. Add vanilla and beat for an additional minute.
Add the flour mixture to the mixer, in batches, alternating with the milk. Mix just until all the ingredients are incorporated, scraping the sides of the bowl when necessary.
Toss mulberries in a tiny bit of flour to keep them from sinking to the bottom of the muffins, then fold them into the batter. Divide the batter between prepared muffin cups.
Make a hole in the top of each muffin and add 1 tablespoon of mulberry jam (or other jam/preserves). Swirl the jam into the batter. Bake for 30 minutes or until a toothpick inserted in the center comes out clean.
Cool muffins for 10 to 15 minutes before removing them from the muffin pan.