Muddle basil leaves, nectarines, and half of the slices of lime in a cocktail shaker or bowl until thoroughly combined.
Slowly stir in simple syrup, tasting as you go, and add more until you have achieved the desired flavor. If your nectarine was very sweet, you may not want to use as much simple syrup in your nectarine cocktail.
Next, add club soda and rum and mix. Pour over ice in cocktail glasses and garnish with the remaining lime slices and a few small basil leaves.