Although "Monsieur Hire" (Leconte) is a great film, I can't really say that its a slice of contemporary French society. I took the title to mean that the question was more on problems/ills associated with modern French society. My first suggestion is "Chocolat" (Denis), but that's more about the residual effects of colonialism. Here are some films that come to mind:
The Dreamlife of Angels (Zoncka) - Unemployment, working class struggle
Rosetta (Dardennes) - Belgian, but more visceral look at poverty and despair than "The Dreamlife of Angels"
L'Argent (Bresson) (or Pickpocket, which is significantly older) - Portrait of materialism and amorality
My Night at Maud's (Rohmer) - The place of religion in contemporary society
La Ceremonie - Bourgeois complacency, class structure between masters and servant
Hope these suggestions help.