Combine 2 tablespoons of olive oil with sesame oil. Brush tuna steaks with oil and season well with salt and pepper.
Lightly coat clean grids of grill with cooking oil spray. Heat grill to medium-high heat. Toss tomatoes with remaining 2 tablespoons of olive oil, lime juice, ginger (if using), salt, and freshly ground pepper to taste.
Brush scallions with oil and arrange in a single layer on grids. Grill 1 minute per side until seared and slightly limp. Transfer scallions to serving plates. Arrange tuna steaks evenly across grids. Cook 3 minutes per side for medium rare (4 minutes per side for medium; 5 minutes for well-done). Turn and cook on the other side to the desired degree of doneness. Transfer steaks to plates, placing tuna on top of scallions. Spoon tomato salad over steaks and serve immediately.