Open and drain the two 5-ounce cans of water-packed tuna, then transfer the tuna to a small bowl and flake it with a fork.
Stir in 2 tablespoons mayonnaise until the tuna is evenly combined and creamy.
Slice each 6-inch sub bun in half lengthwise and set both top and bottom pieces aside.
Place about two small scoops of tuna salad on the top half of each bun.
On the bottom half of each bun, layer shredded lettuce, tomato slices, sliced cucumber, green bell pepper, red onion, and black olives.
Add pickles and jalapeƱo slices over the vegetables on each sandwich.
Cut each slice of white American cheese into triangles and place two triangles on top of the vegetables for each sandwich.
If desired, add a squirt or the remaining 2 tablespoons mayonnaise, close the buns, and serve immediately.