Anyway, the problem I was having, along with others, was Odin 3.07/3.09 was failing on Get PIT for Mapping. It would fail every single time. Didn't matter what I did; switch to another USB, another port, change firmware, nothing mattered. But, in my instance, I've so far solved it, but sheer luck it appears.
I downloaded the firmware from SamMobile, using a Samsung ST150F camera USB instead of the stock USB and the blue Dynex cable (which works much better than the stock cable.
Using an old Toshiba Satellite P205D-S7438's upper left USB port, laptop is running Windows 7 32bit.
All Samsung drivers were installed before starting Odin.
So, here's what I did exactly:
1: Download Samsung Switch from Play Store, connect phone while on and functioning.
2: Start Odin 3.07.
3: Load firmware from SamMobile onto PDA.
4: Wait until Odin says "Leave CS" then restart S3 into Download Mode/Odin Mode (Hold Power, Volume Up, and Home key. Release Power key after first vibration).
5: Odin will print "Removed" then "Added" after phone has restarted.
6: Click "Start" then wait.
So far it's working flawlessly as it's "optimizing" apps. We'll give a good test run to make sure that all is well.