Would buying a new phone give me better service in areas with poor coverage?

No it will not. In fact the areas that don’t have strong coverage will try to transmit using older standards that your phone already supports. Concerning wifi as long as it’s not open and uses a strong security standard requiring a passphrase, you should be fine. However, who ever owns the wifi network can see where you are connecting, they can’t see what data you are passing back and forth. Now this requires your employer to identify you from your phone’s MAC address so if you have not shared this with them, there have no way of them knowing who you are. Additionally you can enable private addresses on your iPhone so that it creates a virtual address for the network that can never be traced back to your phone unless you volunteer that information.