The main goals set for this paper are two-fold. The author first aims to establish a two-mode modified Korteweg-de Vries equation (TmKdV). Second, he seeks to conduct a reliable analysis to examine the conditions that will make this newly developed equation give multiple soliton solutions. He shows that multiple soliton solutions exist for specific values of the nonlinearity and dispersion parameters of this equation. The author also derives more exact solutions for other values of these parameters. He will use the simplified Hirota's method [\textit{C. M. Khalique}, ``Solutions and conservation laws of Benjamin-Bona-Mahony-Peregrine equation with power-law and dual power-law nonlinearities'', Pramana 80, 413--427 (2013); the author, Math. Methods Appl. Sci. 39, No. 4, 886--891 (2016; Zbl 1339.35274)], the \(\tanh/\coth\) method to conduct this analysis. Moreover, other distinct ansatz will also be used to derive other solutions, singular and periodic, of distinct physical structures.