Give this a try. It's advertised for the HTC Thunderbolt, but it works on a number of other phones. I saw a couple reviews saying it works on the S4, but the only S3 review said it didn't. With it only being the one I found, hard to say if that's fully true or just an issue with that one person.
https://forums.androidcentral.com/e...etails?id=net.andirc.lteonfoff&token=IN1J6Rj1
If that doesn't work, try going into the dialer and dial *#*#4636#*#* That should bring up a new set of menus (my Bionic titles it "Testing" but supposed to be a valid code on the S3 too). Go into the phone info settings and you'll see a list of various specs. The above app is basically just a one click shortcut to this screen. Below the spec list, you'll see a selection box that lists the current network mode. Whatever it's on for you would be what you need to select to get back to full 4G. One of the other options would be what you need for 3G only. I'm not familiar enough to know which one is right for the S3, so hopefully someone else can fill that in.
If you select the wrong one, it could affect the radios and not switch back to 4G or pick up 3G properly. I've done that by mistake and occasionally it's just hung up. In my case, going into the regular network settings menu from the home screen settings, then turning data off and back on and then rebooting fixes it. Using widgets and toggle apps to turn data off and on doesn't work. In all cases for me when it switches properly, rebooting the phone resets it back to the default 4G.
Good luck.