An elastic bilayer composed of a stiff film bonded to a soft substrate forms wrinkles under compression. While these uniform and periodic wrinkles initially grow in amplitude with applied strain, the onset of secondary bifurcations such as period doubling typically limit the aspect ratio (i.e., amplitude divided by wavelength) of wrinkles that can be achieved. Here, we present a simple strategy that employs a supported bilayer with comparable thicknesses of the film and substrate to achieve wrinkles with higher aspect ratio. We use both experiments and finite element simulations to reveal that at small thickness contrast, period doubling can be delayed, allowing the wrinkles to grow uniformly to high aspect ratio. In addition, we show that the periodic wrinkles can evolve through symmetry breaking and transition to a periodic pattern of ridges with even higher aspect ratio.