Extending length of wire where the spark is produced on a mini tesla coil
The circuit is not a slayer exciter, it is a PLL-driven SSTC.

I expect your spark is going out because this is a PLL-driven coil. It is designed for a very specific frequency. When you extend the output wire you change the resonant frequency of the system and you will probably lose your phase lock. In order to extend the wire you would have to carefully adjust the capacitor and resistor values on the PLL chip (probably a 4046) in order to move the frequency capture/lock range back to where it needs to be.

In other words, you can't simply extend the wire, and unless you're well-versed in phase-locked loops and how this particular circuit works, and how to modify it to meet your needs, you will not be able to make this change.