I couldn't find anything in the SQLite docs regarding a hash() function. You may be using SQLite with some basic third-party C/C++ plugins included, or you may be using the PHP hash() function (which allows you to specify the hashing algorithm to use).
Generally, a one-way hash is a good way to store passwords, and if it's the PHP hash, used with SHA256, I see no reason why it wouldn't work for you. Just be aware that some hashing functions have demonstrated flaws; MD5 can produce predictable and exploitable collisions, and SHA1 also has theoretical vulnerabilities. SHA2 algorithms including SHA256 are based on SHA1, but have not yet been shown to suffer the same weakness.
In any case, to help ensure a unique hash, use a hashing algorithm that produces a hash equal or larger than the message; it is trivial to prove that, given a finite set of hash values, there cannot be a unique hash value for each of the set of messages larger than the hash size.