The extrapolation technique has been widely used to accelerate the forward-reflected-backward method for monotone inclusion problems. This paper considers a new forward-reflected-backward method with extrapolation ( \(\mathrm{{FRB}_{e}}\) ), which adapts a new extrapolation direction different from the existing acceleration method and uses the latest extrapolation point for the Lipschitz operator. Further, to improve the numerical performance, we propose the linesearch procedure based on the \(\mathrm{{FRB}_{e}}\) by using only the locally Lipschitz constant. Compared to existing methods, our proposed methods not only cover some classical methods, but can also offer a larger stepsize that does not depend on the global Lipschitz constant. We establish the weak convergence of the proposed methods under mild and standard assumptions. In addition, we conduct some numerical experiments on the lasso problem and the \({\ell}_{1}\) regularized logistic regression problem to demonstrate the advantage of the proposed methods.