Be sure to thoroughly muddle the 2 mint sprigs with the simple syrup before shaking.
Muddle simple syrup and 2 mint sprigs in a cocktail shaker. Add rum and lime juice. Fill shaker with ice, cover, and shake vigorously until outside of shaker is very cold, about 20 seconds.
Strain cocktail through a Hawthorne strainer or a slotted spoon into a tall Collins glass filled with ice. Top off with club soda; garnish with more mint.
¾ ounce simple syrup
2 sprigs mint, plus more for serving
2 ounces white rum
¾ ounce fresh lime juice
Club soda (for serving)