We introduce the concept of exponentially $s$-convexity in the second sense on a time scale interval.
We prove among other things that if $f:[a,b]\to\mathbb{R}$ is an exponentially $s$-convex function, then
\[
\frac{1}{b-a}\int_a^b f(t)\Delta t
\leq
\frac{f(a)}{e_{\beta}(a,x_0)(b-a)^{2s}\left(h_2(a,b)\right)^s}
+\frac{f(b)}{e_{\beta}(b,x_0)(b-a)^{2s}\left(h_2(b,a)\right)^s},
\]
where $\beta$ is a positively regressive function. By considering special cases of our time scale, one can derive loads of interesting new inequalities. The results obtained herein are novel to best of our knowledge and they complement existing results in the literature.