[Sprint Samsung Galaxy S III] Dees_Troy:
First off, this will not let you convert a 16GB phone to a 32GB phone. If you try this on a phone that really is a 16GB device, you run a large change of bricking the device and requiring a jtag. These instructions and the files linked are only intended to be used on a Sprint SGS3.

I recently received a Sprint SGS3 32GB that was flashed as if it was a 16GB SGS3. The sticker under the battery said it was a 32GB model and I confirmed that it really did have a 32GB emmc chip inside by using cat /proc/partitions which showed:

cat /proc/partitions
major minor #blocks name

179 0 30535680 mmcblk0

A 16GB model will look like this:

179 0 15388672 mmcblk0

However, because mine was flashed wrong, I only had 12GB of storage instead of 26GB. To correct this problem, you will need a few files:

32 GB pit file and linked just in case someone needs it: 16GB pit file

A full copy of the stock firmware. The one I used is here.

Odin 3.07

USB drivers

You'll also want to back up everything to your removable sdcard. This process will require you to format your entire /data partition which includes all your apps and anything on your "internal sdcard" which includes pictures, backups, etc.

Make sure that you have all of the above files downloaded and have installed the USB drivers. You'll need to extract Odin and the .tar file from inside the firmware zip to known locations on your computer. Make sure your battery has a good charge. Lastly, I take no responsibility if something goes wrong. Also this will *not* trip the flash counter. You're flashing signed Samsung images that do not count towards your flash counter.

Power off your phone.

Hold down volume down and the home button then press and hold power until a screen shows up on your phone.

Release the buttons and press volume up to put your phone in download mode.

Open Odin on your PC

Connect your phone to your computer.

A COM box should light up and Odin should say something like "Attached" after your phone is detected.

Click on the PIT button and browse to and select the 32GB pit file.

Click the PDA button and brose to and select the firmware file. After you select the firmware file, Odin may verify the md5 which may take a few moments.

Click on start and go grab a soda. It will take upwards of 10 minutes to fully flash your device.

Cross your fingers and knock on wood and hope that the phone reboots successfully. On first boot it may boot to stock recovery to wipe cache.

When it boots to Android you probably still only have 12GB of storage, but that's (probably) ok. We'll fix it now.

Go into Settings -> Backup & Reset -> Factory Reset and choose to reset your phone.

Your phone will reboot to stock recovery and perform a full format of all of your data then reboot.

When finished rebooting, you should now have ~26GB of internal storage. Congrats.

Note, if you have trouble getting your phone to be detected by Odin, try a different USB cable and USB port.

I've confirmed that everything works on my phone after doing this. I've been able to activate, do PRL and profile updates, get 3G, 4G LTE, SMS, wifi, bluetooth, NFC, front and rear cameras, etc are all still working.