In a blender, whirl cucumber chunks, 1/2 cup lime juice, the chile, 1/2 cup simple syrup, the salt, and about 1 1/2 cups water until very smooth. Pour into a pitcher and stir in 4 1/2 cups more water. Taste and add more lime juice, simple syrup, and water if you like. Chill until cold, at least 2 hours.
Stir and serve over ice, with cucumber and lime slices.
MAKE AHEAD Through step 1, chilled, up to 3 days.