Samsung Galaxy S3 has an external SD card slot, which is mounted to /mnt/extSdCard
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How can I get this path by something like Environment.getExter
You can use something like - Context.getExternalCacheDirs() or Context.getExternalFilesDirs() or Context.getObbDirs(). They give application specific directories in all external storage devices where the application can store its files.
So something like this - Context.getExternalCacheDirs()[i].getParentFile().getParentFile().getParentFile().getParent() can get you the root path of external storage devices.
I know these commands are for a different purpose but other answers didn't work for me.
This link gave me good pointers - https://possiblemobile.com/2014/03/android-external-storage/