Lightly coat a clean grill with the cooking oil spray. Heat grill to medium-high.
In a small bowl, combine 2 tablespoons of the olive oil with sesame oil (if using). Brush the tuna with the oil mixture. Season with salt and pepper.
Toss the tomatoes with the remaining 2 tablespoons olive oil, lime juice, and ginger (if using). Season with salt and pepper.
Brush the scallions with the oil mixture and arrange on grill in a single layer. Grill 1 minute per side until seared and slightly limp. Transfer to plates.
Arrange the tuna steaks evenly across grill. Cook 3 minutes per side for medium rare (4 minutes for medium, 5 for well-done).
Transfer the steaks to plates on top of the scallions. Spoon the tomato salad over the steaks and serve immediately.