You will only get a USCC 4G signal in USCC coverage areas that have 4G. Currently, there is no sharing/recognition of 4G towers among carriers, like there is with the 3G signal. AT&T can't use USCC 4G and USCC can't use AT&T 4G, etc. It took several years for the 3G signal to be shared among various carriers and it will be the same for 4G.
As far as the strength of the GSIII to pull in a strong signal compared to other phones. Well, the GSIII has had some issues with that, as has been posted in this forum, as well as many other forums. If you are not in a USCC 4G area you might be better off to change your settings to keep your phone only on CDMA. (Menu/Settings/More settings/Mobile networks/Network mode/CDMA.) Your phone won't have to work so hard to keep a 3G signal.
Also, do a *228 call in your Home area. This will update tower information and recognition for your GSIII. Usually that will mean better reception in your Home area and when you travel out of your home area. By all means do call USCC Tech support if you continue to have reception issues.