Joystick button difficult to press in all directions

I’ve got two flotilla joysticks and, while everything works, the button is very tricky to press when the joystick is pushed to the ‘right side’ (away from the button).
My kids were complaining, so I had a look.
Having the button on one side doesn’t seem the best option, although I can see it’s simply pressing down on the ‘shaft’ sticking out of the pots.
I don’t see how you can apply enough pressure at slanted angles to get the button pressed. If you did manage to press it, then it wouldn’t be a reliable press.
The button is only easy to press when the joystick is in the centre position.
I think I can easily solder another pushbutton underneath the board, in parallel over the solder joints of the original button, which you could then press from underneath regardless of the joystick direction. Finger orientation might be an issue!

For these types of joystick- basically identical to the ones used in the Xbox, Playstation, etc- the ability to be able to press in a direction without fear of accidentally pressing the stick button is an intentional feature- to avoid you triggering whatever action the stick is assigned to accidentally when you’re frantically changing direction. It’s not quite so useful on Flotilla, but we’re at the behest of what joystick parts are available :( These are pretty much universal, and thus available in quantity for good prices.

Soldering another push button underneath, or on some short wires, should work well!

OK, point taken, unwanted button presses can just be as annoying as ‘fail’ presses.
The flotilla joysticks work well, and they are analogue, which is much better for many positional things than digital up, down, left, right ‘key-presses’.
PlayStation and Xbox controllers do however have a myriad of other buttons/controls to press.
Although, showing my age, I’m more of a kempston joystick man myself, which is why a simple joystick and button setup is all I was planning in my flotilla ‘roguelike’ ;)