Let $\mathbb{R}^\nu\ni\xi\to\Sigma_{\mathrm{ess}}(\xi)$ denote the bottom of the essential spectrum for the fiber Hamiltonians of the translation invariant massive Nelson model, which describes a $\nu$-dimensional electron linearly coupled to a scalar massive radiation field. We prove that, away from a locally finite set, $\Sigma_{\mathrm{ess}}$ is an analytic function of total momentum.