How could the two-party system be changed to allow for more competition?

Hello there.

How could the two-party system be changed to allow for more competition?

the nature of the electorate itself

What needs to change is either the nature of the electorate itself or, more realistically, the fundamental structure of your national legislature and, to a far lesser degree, the executive. Shift the Legislature to a single voting body with two kinds of seats: upper and lower.