The initial roll-out was blighted by users complaining of rapid battery depletion, freezing displays and, in some cases, even a completely bricked handset.
Samsung has apologised for the cock-ups and has assured S3 users that the problems have been ironed out.
In a statement, the company said: "The fix for the issues with Galaxy S3 Jelly Bean 4.3 upgrade has begun rolling out to selected users in the UK, and will continue to do so. We are sorry for the inconvenience this has caused."
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The software still won't be available to all S3 users as some will need to wait for their mobile network to approve the upgrade.
"Specific upgrade schedules will vary by mobile operators. Please check your phone for the upgrade," the company added.
If there was ever a case for for Android fans to simply buy Nexus phones for timely, stress free software updates than Samsung has done a pretty good job of making it during this debacle.