I rooted my Verizon S3 and now have T-Mobile service but only get Edge connection speed... Sure could use some help getting 3g or 4g speed.
May want to post your question here http://forums.androidcentral.com/t-mobile-galaxy-s-iii-rooting-roms-hacks/
I'm not sure, but you may need radio files specifically for T-Mobile. This assumes that you have a ROM for T-Mobile and that if there's Verizon specific hardware that it will work for T-Mobile.