Estimates for the constant in a Jackson type inequality for periodic functions

New estimates are established for the constant $J$ in the Jackson type inequality
\begin{align*} &E_{n}(f) \leq \frac{J(m, r, \tau)}{n^{r}}\omega_{m}(f^{(r)}, \tau/n). \end{align*}
They improve previously known estimates in the case where $m \to +\infty$, $r \in \mathbb{N}$, $\tau \geq \pi$. Here $f$ is a $2\pi$-periodic continuous function, $E_{n}$ is the best approximation by trigonometric polynomials of order less than $n$, $\omega_{m}$ is the modulus of continuity of order $m$.