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

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.

Hello there.

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

the nature of the electorate itself