Try for June. Southern Europe is less crowded (still shoulder season in many areas, so hotels are open for the year but not so expensive). Later it will also be hot and humid.

You could combine the Riviera area with Provence for more sightseeing or simply take the coastal train toward Marseille and go there, or Aix-en-Provence (stunning smaller city with lots of history and cultureand a couple very nice historic hotels in the center), or Aivignon (church history, also a smaller city where the center inside the walls is walkable), les Baux (realistically requires a car IMO), or go a bit further north toward the Burgundy wine region, for example Beaunne (spelling?) or further north to where the good good Chablis, Meausualt, Montrachet come from, although again you would want a car for such smaller wine villages.