The aim of this study is to develop a novel sixth-order weighted essentially nonoscillatory (WENO) finite difference scheme. To design new WENO weights, we present two important measurements: a discontinuity detector (at the cell boundary) and a smoothness indicator. The interpolation method is implemented by using exponential polynomials with tension parameters such that they can be tuned to the characteristics of the given data, yielding better approximation near steep gradients without spurious oscillations, compared to the WENO schemes based on algebraic polynomials at lower computational cost. A detailed analysis is performed to verify that the proposed scheme provides the required convergence order of accuracy. Some numerical experiments are presented and compared with other sixth-order WENO schemes to demonstrate the new algorithm's ability.