Place fresh blackberries in the cup of a cocktail shaker, then add basil leaves and simple syrup (or agave). Use a muddler to crush the blackberries and bruise the basil.
Fill the shaker with ice, then add tequila, fresh lime juice, and Cointreau. Secure lid, then shake for 10-12 seconds to aerate the drink.
Run a lime wedge along the rim of a glass, then dip upside down into a shallow bowl of kosher salt. Strain into glass and serve with additional blackberries, basil, and/or lime for garnish.