On-chip formation and manipulation of ultrashort pulses would be of great interest to ultrafast optics and integrated photonics communities. One of key issues is dispersion-assisted nonlinear interactions of broadband frequency components. In this article, we show for the first time that a bilayer waveguide for quasi-TE mode without using a slot produces a quite flat and saddle-shaped dispersion profile. Different from previously reported TE-mode waveguides with flattened dispersion, the proposed waveguide exhibits a greatly simplified structure with no need for a high-aspect-ratio slot and has quite small group delay difference in a wide spectral range with four zero-dispersion wavelengths (ZDWs). We study supercontinuum generation in hybrid dispersion regime for the first time, in which the broadened spectrum covers a bandwidth with all ZDWs. It is found that one can obtain greatly improved spectral flatness in supercontinuum generation, with a power variation as small as 3 dB over a bandwidth of >500 nm. Moreover, the proposed waveguides are particularly suitable for low-distortion pulse propagation over a long distance, which is important for on-chip ultrashort pulse delivery.