Your site specific hash, if leaked into logs and generated using a fixed hashing algorithm, would just be a step of obscurity, a rainbow table generated according to that hashing algorithm would be able to extract the original typed password pretty reliably for use in compromising other accounts held by the individual on other sites.

There are ways to actually add security above SSL but they require a per-session nonce. The most secure versions of these operate on some sort of out-of-band time-scrambling algorithm that's issued to users on an individual basis (usually a dongle) which is basically a nonce that's been pre-arranged a few years in advance, these can (if done correctly) provide all the security of on-demand generated nonces and physical keys combined.

Fixed hashing algorithms add such a negligible amount of security that I'd prefer to avoid them due to the tech debt their existence incurs outweighing their advantage.