The well-known generalised-$\alpha$ method, initially introduced by Janssen et al. (2000), is reformulated and extended to include third and fourth-order accurate versions. Due to the Dahlquist barrier, the higher-order methods are only conditionally stable, yet, it is shown that linear combinations of these higher-order methods can yield a new, second-order accurate, unconditionally stable method, provided that the contribution of each higher-order method remains below a specific critical limit. These new methods enhance accuracy and reduce numerical damping in the low-frequency domain. They also allow for user-controlled high-frequency damping through the parameter $\rho_\infty$. The methods are adaptable and can be applied to solve both first-order problems in fluid dynamics and second-order problems in structural dynamics. Notably, one of the new schemes is found to coincide with Park’s method as a special case which is regarded as particularly suitable for stiff problems. The properties of these methods are thoroughly analysed and tested with a wide range of numerical problems. Their effectiveness is demonstrated across various finite element problems, including fluid flow governed by the incompressible Navier-Stokes equations and stiff linear and nonlinear multi-degree-of-freedom structural systems.