The accuracy and efficiency of numerical methods are hot topics in computational fluid dynamics. In the previous work [J. Comput. Phys. 355 (2018) 385] of Du et al., a two-stage fourth order ($S_2O_4$) numerical scheme for hyperbolic conservation laws is proposed, which is based on dimension-by-dimensional HWENO5 and WENO5 reconstructions and GRP solver, and uses a $S_2O_4$ time-stepping framework. In this paper, we aim to design a new type of $S_2O_4$ finite volume scheme, to further improve the compactness and efficiency of the numerical scheme. We design an improved $S_2O_4$ framework for two-dimensional compressible Euler equations, and develop nonlinear compact Hermite reconstructions to avoid oscillations near discontinuities. The new two-stage fourth order numerical schemes based on the above nonlinear compact Hermite reconstructions and GRP solver are high-order, stable, compact, efficient and essentially non-oscillatory. In addition, the above reconstructions and the corresponding numerical schemes are extended to eighth-order accuracy in space, and can be theoretically extended to any even-order accuracy. Finally, we present a large number of numerical examples to verify the excellent performance of the designed numerical schemes.