Fully working Verizon S3 on T-Mobile
Ive seen a couple thread that claim to get you up and running using a Tmobile Sim card in a Verizon Galaxy S3, none of them worked, either no data connection, dialer crashes, or no signal. I really needed to get this working, and i finally have!
Flash the Jelly Bean factory image as outlined in the link above, make sure to follow through the entire OP, phone will NOT be rooted after flashing so take care to install CWM after rooting in order to Flash SU and unlock the bootloader.
At this point you can stick a Tmobile (Possibly AT&T?) sim in the phone, go to mobile networks and you should get a message saying "Switch to global?" hit OK. You should then have bars. if you dont get the "Switch to global" message click around in mobile networks, or switch it yourself (options are GSM, CDMA, or Global. Not sure if choosing GSM will work, I only confirmed Global)
Once you're all set and rooted on the JB ROM, install TBP and APN manager PRO. Using TBP convert APN Manager pro to a system app (Under backup/restore click APN manager pro and select convert to system app). Open up APN manager pro, Select Insert APN, and plug in your T-Mobile APN info which are readily available on the web (make sure to include the MMS fields if you want MMS to work) I used these:
Restart Phone, go back to mobile networks, should get a message asking to switch to Global, hit yes. Congratulations, you now have a Verzion S3 woking with a T-Mobile SIM! At this point only Edge works, i might mess around a bit more to see if 3g/HSPA+ speeds are possible.
Credit goes to open1your1eyes0 for the wonderful write up explaining how to flash the JB factory image.
Credit goes to lair12 for the original guide to use the S3 as a global phone, didnt work fully for alot of people but is a great starting point.