There are usually few seats available at the minimum mileage level on a given flight. When you have a popular destination such as Europe in June, those seats tend to be snapped up soon after they become available (330-355 days before the flight, depending on the airline). Once they're gone, award seats require more miles - usually a lot more miles. This is, unfortunately, a fact of life. You may get lucky and find something, but the odds are not good.