The spectral-element method (SEM) for simulating wave propagation is widely used with adjoint methods for full-waveform inversion. Typically, SEM is used to compute forward and adjoint wavefields, which is then applied to evaluate the Fréchet derivatives for updating the seismic structural model. The Hessian is rarely computed as the high computational and storage costs, although it can improve the accuracy of the model update and model convergence. Instead the approximate Hessian is determined, which is obtained with less computational effort. We present a method for simultaneously constructing Fréchet and Hessian kernels on the fly, which we call Multi-solver spectral-element and adjoint methods (Multi-SEM). Rather than storing all the wavefields, Multi-SEM is computed on the fly and requires only about a 2-fold computational cost when compared to the computation of Fréchet kernels. Numerical examples demonstrate the functionality of the method and the computer codes are provided with this contribution.