Most of the fics I've read are post-OotP, and I rarely ever read fics that don't take GoF into account at all, so most of the fics I know and like take place during sixth and seventh year. I also practically never read anything non-Harry-centric. Some of the fics I put into part 1 of my favorites of course also would fit in here. But I deliberately put those fics with family and mentorship as they are primarily centered on that, less on Harry's sixth year.
Awakening of a Magus by the-dreamer4.
Post-GoF. Mostly gen. Harry's powers suddenly awaken.
PG-13. WIP.
Come What May by
the_minx_17.
Harry/Draco, Harry/Snape. Post-HBP. Pushed into a corner Severus Snape handles it the only way he knows how… he kills Albus Dumbledore. Harry Potter returns to the castle determined to find the power to kill not only Voldemort but also the man who took his mentor away from him, Severus Snape. Instead, he overhears a damning conversation between two people he trusted. Finally using his innate Slytherin cunning, he digs deeper and finds not everything is as it appears. Will it change his loyalties or just the way he fights?
R. Complete.
Dark Phoenix by SofiaDragon.
After OotP Harry decides to escape from the Dursleys and his watchers. During the escape he suddenly finds he can change into a phoenix. As Snape's quarters are the last place anyone would look for the Gryffindor Golden Boy, he spends his holidays in his phoenix-shape with Snape.
WIP. R.
Dumbledore's Army by BobMin.
Harry/Ginny, Ron/Hermione, Neville/Luna, Remus/Tonks, Snape/OFC. Sixth year. After Harry has two visions of Voldemort attacking Hogsmeade and Hogwarts, he does everything he can to assure that those visions won't ever happen like this. In particular he trains the DA.
Sequel
Harry Potter and the Spiritus Crystalus. Seventh year.
R. Complete.
Harry Potter and the Dimensional Trunk by Ophiuchus.
Harry/Ginny, Ron/Hermione. Sixth year. When Ron and Ginny inherit two linked trunks, they gift one of them to Harry so they can easily visit each other.
Complete. PG.
Harry Potter and Dumbledore's Army by Cat Calls.
In his sixth year, Harry has to come to grips with his lot in life. He's awoken during the summer by horrific visions, forcing him to realize that he must train himself not just physically but mentally as well.
Complete. PG.
WIP-sequel
Harry Potter and the Founder's Sword.
Harry Potter and the Runespoor Shield by Ms Spider.
Minor Harry/Ginny. Harry finds an old trunk his mother has left for him. How did she find the things in it, why did she think he'd need them? And how come Ginny's kitten looks just like the kitten Hagrid brought for his first birthday? And can Harry and Snape deal with their mutual animosity long enough to figure everything out?
Complete. PG-13.
Harry Potter and the Secret Sanctuary by Farfumsane.
Harry/Susan, Harry/SallyAnn Perks, Harry/OFC, and (eventually) Harry/Ginny. Harry and Co’s 6th year is filled with new powers, places, experiences and faces. HighPower!Harry.
WIP. PG-13.
Harry Potter and the Veil of Mystery by Semprini.
Harry/Ginny, Hermione/Neville, Ron/Pansy. Sixth year. After Harry's success with the DA Dumbledore makes him the DADA-teacher during his sixth year.
PG-13. Complete.
Sequels
Harry Potter and the Ring of Reduction Seventh year.
Phoenix Intuition. (This takes place five years later and I didn't like it.)
The Refiner's Fire by Abraxan.
Harry/Ginny, Ron/Hermione, Remus/Tonks. Sixth year. After a rare magical illness Harry becomes outstandingly powerful. He has to deal both with teenage-problems and more importantly with many plots by Voldemort and his Deatheaters to kill him.
R. Complete.
Sequel
The Time of Destiny, ficlets taking place during the epilogue
On Second Thoughts and
A Fox's Tale,
one-shot sequel
A Very Harry Christmas.
Scion of Gryffindor by Crys.
Harry/Tonks. On his sixteenth birthday Harry gets a surprising visitor: the ghost of his ancestor Godric Gryffindor. Harry doesn't quite trust Dumbledore anymore due to everything that happened in OotP. So instead of returning to Hogwarts, he gets training from Tonks, Shacklebolt, Remus and Godric.
WIP. PG-13.
Sunset over Britain by Bobmin.
Harry/Hermione, Draco/Luna, Neville/Ginny. Harry's life at the end of his fifth year was in utter chaos and his life was not going to get any better any time soon. From the flames, a brotherhood thought dead will reemerge, a family will break, trust will be destroyed, an enemy will change their path, a father will finally be found, a love will bring hope, and a hero shall rise from the ashes to become the legend he was born to be.
R. Complete.
WIP-sequel
Sunrise over Britain.
Trinity by Katling.
Harry/Ginny, Ron/Hermione. Harry now knows what he has to do - find and destroy the Horcruxes and Voldemort. Ron and Hermione refuse to leave his side and they choose to become something that the Dark Lord will fear...the Trinity.
PG. Complete.
My other lists.