Grate the cucumber on the large holes of a box grater into a bowl. Toss with 1/4 teaspoon salt and pour into a fine-mesh sieve set over a bowl. Let stand for 10 minutes. Using your hands and working with a bit of the cucumber at a time, squeeze the excess liquid out of the cucumber and place the cucumber in a large bowl. Add the yogurt, feta, garlic, 1/4 cup of the chopped mint, 1/4 teaspoon salt and 1/4 teaspoon pepper to the bowl with the cucumber. Stir to combine and add more salt and pepper to taste. Transfer to a shallow serving bowl.
Toss the tomatoes, red onion, olives and remaining 2 tablespoons mint together in a medium bowl. Sprinkle the mixture over the cucumber mixture. Serve the dip with warm pita or vegetable chips.