A new iteration of Galaxy S III Android 4.1.1 Jelly Bean I9300XXDLI5 leaked ROM is out there which is quite a clean and working featured peace of software. And this latest build includes fixes for about all of issues found in previous release.
The official Android 4.1.1 Jelly Bean for Galaxy S III is just near to release but if you can not wait for that then download Android 4.1.1 Jelly Bean build I9300XXDLI5 for S III from source to give it a try.