I am not a developer and can take no credit for this (and consequently cannot provide support for this)! All credit and thanks go to @k1mu for this method.
For those of you who want root access to your phone and do not want to use Kingo (root without Kingo) here is a safe method. The best part is if it doesn't work there is no harm done to your phone, it simply will not root. This will provide root on the stock software and DOES NOT trip the KNOX flag (your KNOX flag will remain 0x0). The way to trip the KNOX flag is by changing the kernel or bootloader/recovery and this does neither. It may change the System Status to "Custom" (this can be seen in Download Mode or in the phone's menu under Settings->About Phone->Status->Device Status) but there are ways to change that, see the details in the original post.
This does not unlock the boot loader, does not install a custom recovery or allow you to load custom ROMs.
@k1mu created this method using an Experia exploit and modifying it for the Galaxy S4 full size, however it works for many other Android devices. I used this to root a Verizon Galaxy S4 Mini SCH-I435 and other people have reported this working for the S3, Note 2, other carrier phones and some non-Samsung devices.
This will provide root access on your phone and install SuperSU. This works with Android 4.3 as well as 4.2.2.
Here's the basic steps to root your S4 Mini (for the full step-by-step visit the thread):
1. Setup USB drivers on your computer.
2. Enable USB debugging on your phone.
3. Enable USB ADB access.
4. Download and unzip the Saferoot.zip
5. Root your phone
6. Click "Thanks" on k1mu's thread for this (yes I added this step ).
If you have questions or run into problems reply to k1mu's thread. He is very helpful and quick to reply.
If you find yourself needing to unroot, visit post #2 in the original thread and follow the steps. You can find SCH-I435 stock firmware and Verizon software in this thread (post #10 & #12). I HAVE NOT TRIED THESE SO USE AT YOUR OWN DISCRETION.