Gently press the mint leaves at the bottom of a cocktail shaker to release their oils without tearing them. Introduce the white rum, lime juice, simple syrup, and a generous scoop of ice to the shaker. Secure the lid and shake vigorously until well chilled. Strain the mixture into an ice-filled cocktail glass. Complete the drink with a dash of club soda, then embellish with lime slices and sprigs of fresh mint.