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.

acquarello
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