Short answer: He has no legal right to do so. Now for the long answer:
As has been discussed above, Jackson only had the rights to The Lord of the Rings and The Hobbit, and he got the rights with no help from the Tolkien Estate or Christopher Tolkien. It is extremely unlikely that the Estate will ever sell the rights to anything else to Jackson or anyone associated with him, in light of Christopher Tolkien's overwhelmingly negative response to the LotR movies.
The rights to Tolkien's other work are held by his Estate, which is in the UK. Under British law, copyrights on literature expire 70 years after the death of the author. Tolkien died in 1973, so some of his work may become available for adaptation in 2043.
However, the jewel of the Tolkien catalog is undoubtedly The Silmarillion, and this is a special case, quite different from most of Tolkien's work. Why is it different? Because Christopher Tolkien has an authorial credit on The Silmarillion. This means that only he can sell the rights to it until they expire, and they won't expire until 70 years after he dies. He's an old man now, in his 90's I believe, and probably won't be around for very long, unfortunately. But the fact still remains that the rights to the Silmarillion will not expire until 70 years after he dies, which will be 2086 at the very earliest.
It goes without saying that Peter Jackson won't be around in 2086, so if the rights expire before the Estate decides to sell them, Jackson won't be the guy who makes the films.
As I understand it, anyone can buy the rights to reboot the LotR and Hobbit franchises, as long as whoever owns them is willing to sell. But the rights only apply to the material in the books in question (all three Lord of the Rings books as well as The Hobbit), so no one can make a sequel to any of the books.
If you really want to see more films based on Tolkien's work, your only hope is that after Christopher Tolkien dies, the Estate decides to sell the rights more freely than it has done in the past. This is certainly possible, since the Estate will soon be controlled by people who never met Tolkien, and therefore, might not be as emotionally invested in the works. When people who don't know Tolkien, and never did, have their finger on the button, they may well opt to make easy millions by selling everything they can. As Jason Baker mentioned in his comment below, it is also possible that the Tolkien family will come into some sort of financial crisis and be forced to sell everything to pay the bills. I wouldn't count on it, however.
Otherwise, you'll have to settle for remakes of the existing movies, and - beginning in 2043 - a few bits and pieces becoming available for adaptation. The Silmarillion, however, won't be a movie until someone finally sells the rights, or the rights expire in 2086 at the earliest.
For further reference:
You would do well to begin with Jason Baker's excellent answer here.
After that, check out the following links.
This article points out that, with the LotR and Hobbit films now having raked in over $10 BILLION dollars, it is inevitable that we'll see more of Middle-earth on the silver screen, no matter how long it takes. Hollywood doesn't forget successes of this magnitude.