Finally managed to do what I was trying to accomplish on my VZW Galaxy S3 for almost a year.
I'm not sure if anyone else has already accomplished this or not. If anyone has, then sorry about the duplicate post (even though I haven't found any posts specifically describing how to do it).
It took me some research to see that both t-mobile and vzw run on band 4 block f (1700mhz-f), so I thought they should both be able to tune to the same lte band. Even though I was able to get H+ from a t-mobile prepaid sim to work on the vzw note 3, I was still not completely satisfied, so I kept at it till lte finally worked too.
I do give credit to the thread about "All things service mode" which described how to enable the service mode menu and then how to turn on lte band 4.
This DOES confirm that by turning on lte band 4 under service menu, it does really turn on, no firmware update is required to enable the additional layer of the lte band.
It's almost 4 a.m. so I'll get to the details a little later, but just wanted to post this as soon as I got it working, I know it'll make lots of people happy.
I got 28 mbps down and 6 mbps up on t-mobile where I get about 8m and 2m with vzw lte, respectively.
Look at the screen shots below. I did a little research and was able to get that annoying "non-Verizon sim" warning off too, but credit for that goes to someone else here on xda, I didn't figure that out, I will just relay that here for a one stop shop on how to use a t-mobile or other GSM sims in the vzw version. I will link to that post later when I have more time. Anyway, see below and let me know of your thoughts, I will add more details later.
Do this at your own risk, if any anything goes wrong, don't blame me for it. Make a nandroid backup before you start. To get 4G lte to work on t-mobile, and to remove the continuous sim warning, you'll have to be rooted. Just to get 4G hspa+ to work, you don't need root, but you'll have the annoying sim reminder/warning in your notification all the time.
- Go to settings>more settings>mobile networks>network mode and change to "Global" mode. Also turn off your mobile data at this point.
- Turn off your phone and replace your vzw sim w/ a t-mobile sim with prepaid or postpaid 4G service. Turn the phone back on. When it has finished booting up, you should have basic gsm service on t-mobile, you'll get a warning about a non-Verizon sim, don't worry about that right now, it won't hurt anything.
- Now if you want t-mobile 4G lte and you have root, follow the thread for All things "ServiceMode" related to turn on lte band 4 through service mode. Restart the phone after enabling lte band 4.
- Go to "mobile networks>access point names" under more settings, click the "+" on the bottom to add an apn, call it T-Mobile LTE and type "fast.t-mobile.com" without the quotes under APN. Leave everything else blank (you can add some of the other stuff later if you want to get picture messaging to work). Click the menu button and hit save. Once you're out, make sure you click on the green radial button to select the new APN you just created.
- Now exit settings and turn on your mobile data. Give it a minute. You should get the 4G lte symbol after a little bit. You may have to restart the phone once for it to work.
If you're not in a good t-mobile 4G lte coverage area and just want the traditional hspa+ option, follow the same steps, but DO NOT enable lte band 4, and instead use "epc.tmobile.com" for apn. This way you won't need to be rooted. If you are already rooted and did enable lte B4, just switch the phone to gsm/umts mode under network mode, that will disable lte.
Now if you want to go the extra mile and remove the sim warning, get Titanium Backup from Play Store and freeze the app "Setup Wizard", the app with the "magic wand" above the red "check mark". You'll have to be rooted to do that.
If you follow similar steps, you can also get AT&T hspa+ to work, obviously with an AT&T sim and AT&T apn settings ("pta" for the apn, but AT&T has their lte on a different band, so you can't get AT&T 4G lte to work on this phone. For their hspa+, the steps are the same as for t-mobile hspa+, except for the actual apn settings.