Add the lime wedges and ginger simple syrup to a highball glass with a sturdy bottom. Muddle until the lime breaks down. Add the fresh mint leaves and lightly muddle a few times.
Pour in the white rum and add a handful of ice. Stir to both combine and chill. A bar spoon works well for this, but you can use a regular kitchen spoon if you don't have one.
Top with chilled club soda and gently stir to combine. You don't want to stir too forcefully, or you can lose that delightful carbonation.
Place the fresh mint sprig in the palm of your hand. Quickly clap over it to release its fragrance and garnish.