Time Attack Mode is a convenient way to improve your records in the game itself.
That same save file will have the green symbol. To obtain one, simply beat the game with a saved file. Getting a New Game+ File is very convenient for seeing the player's current in-game time while pausing, or using its checklist of expansions in the map screen. A New Game+ File is identified on the file select screen by a green screw attack symbol on the right side.
