To start off, any configurations that involve the cut portion of the light strip, are not officially supported by TP-Link and there can be no guarantees that a setup will function, or communicate with the Kasa application correctly.
The KL420L5 makes this a little more complicated, and I'm not sure it is possible due to the RGBIC features of the strip. After cutting the strip, you will have to look into third party connectors that attach to the ends of the cut strip and would allow you to add a segment. These connectors vary by Pin Number, Order of Pins, and What type of coating is on the strip, so make sure you research what adapters would give the best chance of working with your strip.
If you are looking at connecting another strip to the end of your existing KL420, this will most likely not work, as the controller is only rated to provide power fox x number of lights, and adding another strip may overload this. Also, the Kasa Application has no controls for Independent Control Lights that are beyond the set length, or at least so I would expect.