I've made a little tool to serve your documentation and keep it up to date: https://github.com/derekdreery/cargo-docserve. It builds your docs, serves them from a port of your choosing (8000 by default), and rebuilds them when the source changes.
There are some limitations (for example it just watches "src", so it ignores cargo-rerun-if-changed in build.rs). For now I'm not publishing it, but if I get positive feedback I will.
I would be happy to do a PR for cargo, but I wanted to see what other people said first. If it was part of cargo it could use the internal api which would allow more things. A potential reason for not making it part of cargo is because it pulls in a webserver as a dependency so would make the binary larger potentially