Samsung has officially declared no way out for any further official update via twitter. Galaxy S3 is now 3 year older, and its no surprise that it will receive Marshmallow update either. However there are plenty of Custom Rom available in the market.
Unofficial Marshmallow Custom is available on XDA Developer Forum but the stability is a matter of concern.
Surprisingly Cyanogenmod has started rolling out Marshmallow nightly builds. Earlier Cyanogenmod community was believed to have stopped working on Galaxy S3 custom roms. The last build were CM 11 nightly builds- the Kitkat version. Now Cyanogenmod has skipped Lollipop version and are now working on Marshmallow Custom Rom.
If your are really enthusiastic for your beloved phone to taste Marshmallow, give this a try. It may consists of bug, but nothing to worry, since Cyanogenmod are realising builds every day. You may opt to wait for the final Stable build.
Want to give a try?
Caution: Use at your own risk.
For unofficial custom rom, follow: http://forum.xda-developers.com/galaxy-s3