There are a couple of things to do in Paris that are often listed as "beyond the obvious," but I see them listed so frequently they may be more properly listed among the obvious now. At any rate, one of these is the tour of the catacombs cut under the streets of Paris -- an endless maze of tunnels occasionally filled with stacks of human bones, disassembled and stacked in neat patterns. There's a set route that's easy to follow, but if you wander off in an un-lit direction, you could be lost for a very long time.
Another is the tour of the sewers. I haven't tried this myself, as it seems a bit too much along the lines of "They're tourists. They'll buy anything." But people do tend to report that it's worth the effort. As I understand it, some of the "charm" is connected to the romance of the sewer-chase scene in Les Miserables.
And the third is the Pere Lachaise cemetery, where many famous people are buried, including Jim Morrison.
Aside from those rather commonly mentioned second (or third) tier attractions, have you considered any of these:
1) The Musee Marmottan (full of paintings by Monet beyond those you've probably already seen at the Musee d'Orsay).
2) The Orangerie -- just across the river from the Orsay, a relatively small museum that houses Monet's huge, panoramic water lilies paintings. If you've only seen his smaller paintings (the size you might be able to get into your house), you should not miss these.
3) The Rodin museum.
4) The museum of the history of Paris (I believe it's called the Carnavalet, or something similar) -- pretty interesting, but best if you can read French.
5) The base of the old Louvre fortress, accessible from the basement of the new Louvre museum
6) The Medieval museum, in the Latin Quarter
7) The Roman amphitheater, near the Medieval museum
8) The Eiffel Tower (oops, nevermind)
9) If your trip is in cold weather, go ice skating in front of the Hotel de Ville (town hall, not a hotel)
10) On Sunday morning, attend 11:00 services at the St. Sulpice church, and stay for a 15-20 minute organ concert immediately following the service. This church is where Charles-Marie Widor was concertmaster, and the spirit of his music is retained in the sound of the organ and the selection of music played in these concerts (if you're not familiar with Widor, find someone who will lend you a recording of the Toccatta from his 5th symphony, commonly known as "The Widor Toccatta." And if you're asking your friends if they have a recording you can borrow, remember to pronounce "Widor" like "Veedore.")
11) The Conciergerie -- the prison that held many of those who were beheaded during the French Revolution(s), including Marie Antoinette's cell.
12) Any of several flea markets or the flower market or the bird market.
13) And if you want a tour of the Louvre and/or Orsay museums that will change the way you look at art forever, fly me over, with my wife, first class, all expenses paid, and I'll be happy to provide it.
Now, you also mentioned that you're looking at other parts of France. Most Americans, at least, first consider the Normandy beaches, Mont St. Michel, the Loire, and Provence when looking for French destinations besides Paris. But let me suggest you also consider the Dordogne region. The Dordogne is a river that runs from south central France westward to the Atlantic at Bordeaux (after becoming the Gironde River). The most captivating part of the region is in the river's middle section, where it flows through a beautiful landscape of rolling hills peppered with charming small towns and castles. The roads are all small and nothing moves fast in this region. During the Hundred Years War, the Dordogne sporadically formed the border between English and French lands, so there are medieval castles on every cliff and seemingly at every bend in the road. Many of the villages retain their medieval character, as well. At Les Eyzies, there's the famous Lascaux cave paintings. Many other examples of prehistoric art can also be visited in the region. Prices are quite moderate in the region for accommodations. It's wonderful to stay in a castle and spend the day in a canoe floating down the Dordogne, passing medieval towns and castles at every bend, stopping at a beach along the way for your "pique-nique" of a loaf of bread, a jug of wine, and, well, whatever else pleases your fancy. And most of the Dordogne region is coincident with the Perigord, from which we get Perigord black truffles, foie gras, cassoulet, and other wonderful French "home cooking."
Does any of that sound appealing?