You can fly to Nice with practically any airline that has hub in Europe. KLM=amsterdam, using Continental/NW miles (or xfer from Amex points), UA=London Heathrow where its partner British Midland has hub, UA=Frankfurt where its partner Lufthansa has hub, etc. The key is finding a mileage seat to Nice in May, which is peak season b/c Cannes Film Festival and other Grand Prix races in Nice/Monaco.

If you have trouble finding free seat from an European hub to Nice, but can find seat from USA to European hub, also consider buying a RT tkt to Nice.