Okay, I have several recommendations to add. I think that several of these stories have probably been mentioned in this thread already, but no good fanfiction can have too many recommendations
First, I have come across not one, not two, but three really good fics that give alternate POV's for New Moon (a la Midnight Sun). They are:
Dark Side of the Moon, by blondie AKA robin;
Thy Beauty, bu subtlynice; and
Ithaca is Gorges, bu giselle-lx. The first two are from Edward's POV, the third alternates between Edward and Carlisle. All three have that most wonderful of qualities--I keep going back to reread them.
Each of the three authors mentioned above has also written other stories that have gone into my "favorites" list. blondie AKA robin has written "A Dickens of a Dream" and "Brotherhood." The first puts Bella into a "Christmas Carol" scenario, and tells how a group of "ghostly" visitors one Christmas Eve inspire her to move to Forks to live with Charlie. The second tells of a hunting trip that Edward, Emmett and Jasper take, during which they can't stop quoting, respectively, Shakespeare, Monty Python and The Princess Bride.
Subtlynice has written a pair of nice vignettes from Angela's POV. They are titled "Optimism" and "Bravery" and both take place during New Moon. The first is just after Bella emerges from her "zombie" period; the second takes place on the day that Bella, Edward and Alice return to Forks High after the events in Volterra. Angela fans (I'm one) will want to check these out.
giselle-lx has written a nice little story called "Form 1040," about Carlisle working on paying the Cullen family taxes. It will come as no surprise to people that vampires have to deal with some interesting issues when it comes to the IRS.
One other story that I've really come to like is
Finding Twilight, by Mrs.Alice.Whitlock. This one starts when Alice and Jasper, on a shopping trip (imagine that!), wander into a bookstore and find a set of four books. When they get home, the family sits down to read them together. The opening words of the first book are "I'd never given much thought to how I would die..." I'm having a lot of fun reading this one.
I promise they're good!