First of all, every stored password should be hashed with a different pseudo-random salt.  Second, SHA-256 is not appropriate for storing passwords; instead, you want to use a key stretching algorithm, as has already been mentioned.  There is a lot more detail at Crackstation.

An encrypted hash is also called a keyed hash, and the key is sometimes called a "pepper."  There need not be a separate encryption step.  Instead, the secret key is part of the input to the hash function.  That can improve security somewhat.  There are two ways attackers can get a copy of your password hashes.  One is to compromise the storage mechanism, as with SQL injection.  In that case, a keyed hash can make it effectively impossible to retrieve plaintext passwords from the hashes because the key can be compiled in to a program or otherwise kept outside the password storage mechanism.  The other way is to compromise the OS itself.  In that case, the key is compromised, and is no longer an impediment against attempts to attack the passwords.

So, the short answer is yes, but you have to do the underlying work right, first.

Are there any disadvantages to encrypting the password hash?
In episode 345 of Naruto Shippuden (オレは地獄に居る/I'm in hell), when Obito arrives on the battle field, he sees how Kakashi killed Rin. The wiki summarizes this scene as


  On his way to the battle site, Obito has a vision of Rin through his empty eye-socket. He arrives just in time to see Kakashi run his Chidori through Rin's chest. Rin's death awakens both their Mangekyō Sharingan.


In the next scene we see that the Kiri-nin are clearly disappointed by the fact that Kakashi killed Rin, one of them says "Damn it! He did it!", and another replies "After all the trouble we went through to get her!"

But I don't understand why he had to kill her, why they couldn't wait, e.g., for reinforcements to arrive. Is there some good reason why it was better, for her, or for the village, that she dies, rather than to be captured by the Kiri-nin?
