Tempered fractional derivatives and the corresponding tempered fractional differential equations have played a key role in physical science. In this paper, for solving the tempered fractional ordinary differential equation, the predictor–corrector (PC) methods with uniform and non-uniform meshes of Deng et al. (Numer Algorithms 74(3):717–754, 2017) are developed, by using the piecewise quadratic interpolation polynomial. The error bounds of proposed predictor–corrector schemes with uniform and equidistributing meshes are obtained. We proved that the presented numerical method has a higher-order convergence order $$O(h^3)$$. Also, some numerical examples are constructed to demonstrate the efficacy and usefulness of the numerical methods. Finally, the results of PC schemes with uniform and non-uniform given in Deng et al. (2017) and presented schemes (improved PC with uniform and non-uniform meshes) are compared for different values of parameters.