I have no idea about Amazon phones. But if it has developer mode then enable it and use ADB
http://developer.android.com/guide/developing/tools/adb.html
Open Command Prompt where ADB is located.
use command: "adb shell pm list packages"
It will list all installed packages(apps) on your phone, notice package name of your app and use cmd
"adb shell pm path com.example.appyouneed"
It will give you package path then use cmd
"adb pull /data/app/com.example.appyouneed-1/base.apk"
replace path /data/app by whatever path you get for apk file, apk will be copied where adb is located, transfer to S3 and install or as you want by cmd prompt use:
"adb install app.apk"
making sure s3 is in developer mode as well.