Bring 3 cups currants, the sugar, wine, and salt to a boil in a medium saucepan over medium-high heat. Reduce heat, and simmer until berries are soft, about 1 minute. Puree in a blender. Pour through a fine sieve into a bowl (you should have about 1 3/4 cups).
Return strained puree to pan; bring to a boil. Add remaining currants, return to a boil, and skim if necessary. Mix cornstarch with 1/4 cup water until cornstarch dissolves, then whisk into currants. Cook over high heat, stirring occasionally, until boiling in center and mixture looks clear, about 5 minutes. Let cool 5 minutes.
Divide mixture between 6 small glasses. Chill until set, at least 1 1/2 hours and up to 8 hours. Dollop whipped cream on top, and serve with fresh currants.