An adaptive even order weighted essentially non-oscillatory polynomial reconstruction procedure (WPR) with Z-type weights (WENO-Za) is proposed to reduce the numerical dissipation and improve the resolution of the fine-scale structures in a long-time simulation of the hyperbolic conservation laws. The new spatial solver combines the even and odd order (even order plus one) WENO finite difference schemes together. The novel WENO-Za scheme adapts the ideal weights, instead of the nonlinear weights, between the even and odd order based on a nonlinear transition function, which is a function of the odd order global smoothness indicator and a user-adjustable parameter. The dissipation and dispersion behaviors of the WENO-Za scheme is illustrated by the approximate dispersion relation technique. For the Euler equations, the WPR procedure is performed on each positive and negative split flux variable in the characteristic space. The performance of the WENO-Za scheme, in terms of resolution, essentially non-oscillatory shock-capturing, and efficiency, are evaluated by solving several benchmark shock-tube problems, such as the extended shock-entropy interaction problem, the extended shock-density interaction problem, the two blast-waves interaction problem, the Riemann problem, and the forward-facing step problem. The results show that the WENO-Za scheme can 1) retain the fine-scale structures in the smooth regions better due to the reduction of numerical dissipation of the even order scheme in exchange for a slightly larger dispersive error in a long-time simulation and 2) capture localized strong gradients and shocks in an essentially non-oscillatory manner due to the strong sub-stencil biasing in the odd order scheme. The space-time evolutions of the transition functions demonstrate the robust adaptive nature of the WENO-Z4a5 scheme in the regions of shock, contact discontinuity, and rarefaction waves.