A class of finite difference Hermite radial basis functions weighted essentially  non-oscillatory (HRWENO) methods for solving conservation laws was presented by Abedian (Int. J. Numer. Meth. Fluids, 94 (2022), pp. 583--607).To reconstruct the fluxes in HRWENO, the common practice of reconstructing the flux functions was employed. In this follow-up research work, an alternative formulation to reconstruct the numerical fluxes is considered. First, the solution and its derivatives are directly employed to interpolate point values at interfaces of computational cells. Afterwards, the point values at interface of cell in building block are considered to obtain numerical fluxes. In this framework, arbitrary monotone fluxes can be employed, while in HRWENO the classical practice of reconstructing flux functions can be considered only to smooth flux splitting. Also, in the process of reconstruction these type of schemes consider the effectively narrower stencil of HRWENO methods. Extensive test cases such as Euler equations of compressible gas dynamics are considered to show the good performance of the methods.