We study tropical Tevelev degrees arising from maps between certain tropical moduli spaces of curves. Building on work of Dawson and Cavalieri, who defined and computed tropical Tevelev degrees in the case of degree $d = g+1$ and $n = g+3$ marked points, we extend the theory by introducing an additional integer parameter $\ell$. In our framework the curve degree and number of marked points vary as $d = g + 1 + \ell$ and $n = g + 3 + 2\ell$, and we analyze the resulting tropical Tevelev degrees for both positive and negative values of $\ell$. This tropicalizes results of Cela, Pandharipande, and Schmitt on algebraic Tevelev degrees. We then further broaden the framework by introducing generalized tropical Tevelev degrees, providing the tropical counterpart to the generalized Tevelev degrees studied by Cela and Lian. These results establish a wider set of computational and structural patterns for intersection calculations on tropical moduli spaces and reveal new behavior beyond the classical setting.