This paper presents an efficient strategy for constructing Reduced-Order Model (ROM) bases using Taylor polynomial expansions and Fr{é}chet derivatives with respect to model parameters. The proposed approach enables the construction of ROM bases with minimal additional computational cost. By exploiting Fr{é}chet derivatives -solution to the same problem with distinct right-hand sides -the method introduces a streamlined multiple-right-hand-side (RHS) strategy for ROM bases construction. This approach not only reduces overall computational expenses but also improves accuracy during model parameter updates. Numerical experiments on a two-dimensional wave problem demonstrate significant efficiency gains and enhanced performance, highlighting the potential of the proposed method to advance computational cost-effectiveness, particularly in seismic inversion applications.