Preheat the oven to 425 degrees. Mix 1 cup of the swiss cheese with the string cheese.
Lightly brush a baking sheet with olive oil. Unroll the dough, cut in half to make two 6-by-9-inch pieces and place on the prepared pan.
Spread half of the mixed cheese along one long side of each rectangle, leaving an inch border between the cheese and dough edge. Top each mound of cheese with half of the basil, a quarter of the olives, 1/4 cup tomato sauce and a third of the mortadella. Fold the dough over the filling lengthwise, pressing the edges to seal. Make slits in the top of the dough to allow steam to escape during baking.
Bake the stromboli until brown, about 15 minutes; remove from the oven. Brush each with olive oil, then scatter the remaining olives, mortadella and cheese on top. Return to the oven and bake until the cheese melts and the crust is golden. Rest for 5 minutes before slicing.