Here we go for Note 3 File Manager for our Galaxy Tab 3.
All credits goes to @android.zealot.fervor in his original thread.
I've just modded it a little as screenshots below:
It is a flashable zip, install through recommended last philz recovery(click on image in my signature)
Just in case before flashing it, use a root explorer, go to system/app and rename SecTabletMyFiles.apk in SecTabletMyFiles.bpk, then reboot and flash. If encounter Fc clear data in app settings and try again.
It is tested on my Tab 3 8", so should work even on others