There seems to be a misunderstanding here: There is no Appendix or unfinished tales that give any sort story for what happened in the 60 yrs between the Hobbit and the Lord of the Rings. Other than a few dates (i.e. so-and-so was born on this day) and couple of two-line events (in such and such year a battle happened between so and so). So there really is no "story" left by Tolkien, finished or otherwise to make into a movie. Or at least not one which wouldn't require a large amount of invention to flesh out, thus making it mainly a fan fiction.
I can only see three possibilities of events to make a movie out of in this time period:
1) The White Council (Gandalf, Saruman, Galadriel, Elrond, et. al) finally putting forth their strength to throw the Necromancer (Sauron) out of Southern Mirkwood. Initially just an excuse to get Gandalf out of the narrative of the Hobbit, it is a nice tie in to events in Rings, allowing some of the big movers and shakers of Middle-eath more screen time. It also ties to the more epic plots in the earlier movies. Saruman was already going bad at this point and agreed to the attack largely to prevent Sauron from finding the Ring before he could. The problem is that this happened simultaneously with the last half of the Hobbit (so it does not really advance the timeline any, as Gandalf is able to make it back for the battle of the five armies at the end of the Hobbit) and not much is detailed about the event.
2) Aragorn comming of age and falling in love with Arwen. This was a major focus of the Rings movies and would be prime fodder to 'link' the two seperate movie franchises. It does advance the timeline, but even less is know about the details of Aragorn's early adventures than about the Attack on Dol Guldor mentioned above. Other than falling in love with Arwen, we know he spends much of his time in the wild on a kind of 'errantry', learning and growing. He spends time in disguise serving in both Rohan and Gondor for Theoden's and Denethor's fathers. That's about it.
3) Balin's failed attempt to re-colonize Moria happens during this period as well. While it does serve as a good link from the dwarf-centric Hobbit to a place visited in Rings, I'm not seeing how they can make a movie about just this and link it in to the overall epic plot of the series.
Finally, even if Christopher Tolkien is sitting on some hitherto unrevealed manuscript relevant to this time period (highly unlikely), the film makers would not have the rights to use any of the material anyway, as most of the material outside of the Hobbit and Rings is off limits.