The present research applies an improved version of the modified Extended Direct Algebraic Method (mEDAM) called $r$+mEDAM to examine soliton phenomena in a notable mathematical model, namely the (2+1)-dimensional Nizhnik-Novikov-Veselov Model (NNVM), which possesses potential applications in exponentially localized structure interactions. The generalized hyperbolic and trigonometric functions are used to disclose a variety of soliton solutions, including kinks, anti-kink, bell-shaped and periodic soliton. Some 3D graphs are plotted for visual representations of these solutions which highlight their adaptability. The results provide a basis for practical usage and expansions to related mathematical models or physical systems. They also expand our understanding of the NNVM's dynamics, providing insights into its behavior and prospective applications.