Given a frequency $\lambda$, we study the Riesz summability of $\lambda$-Dirichlet series $\sum_{n=1}^\infty a_n e^{-\lambda_n s}$ generating holomorphic functions of finite order. We present a new condition on the frequency $\lambda$ ensuring that, for any $k \geq 0$, each $\lambda$-Dirichlet series that is somewhere Riesz summable of some order and admits a holomorphic extension $f$ to the right half-plane $\C_0$ satisfying $f(s) = O(|s|^k)$ as $|s| \to \infty$ on $\C_0$, is in fact Riesz summable of order $k$ on $\C_0$. This extends Bohr's theorem, which corresponds to the case $k = 0$. Our work improves a recent result of Defant and Schoolmann, who showed the above property under Landau's condition (LC). Along the way, we also establish novel optimal bounds on the coefficients of such $\lambda$-Dirichlet series, which may be of independent interest.