We establish criteria of Hille-Nehari type for the half-linear second order dynamic equation ( r(t) Φ(y ^{∆}))^{∆} +p(t) Φ (y ^{σ} )= 0, Φ (u) = |u|^{α-1} sgn u, α \> 1, on time scales, under the condition R$\infty$ r 1/(1-α) (s) ∆s \< $\infty$. As a particular important case we get that there is a (non-improvable) critical oscillation constant which may be different from the one known from the continuous case, and its value depends on the graininess of a time scale and on the coefficient r. Along with the results of the previous paper by the author, which dealt with the condition R$\infty$ r ^{1/(1-α)} (s) ∆s = $\infty$, a quite complete discussion on generalized Hille-Nehari type criteria involving the best possible constants is provided. To prove these criteria, appropriate modifications of the approaches known from the linear case (α = 2) or the continuous case (T = R) cannot be used in a general case, and thus we apply a new method. As applications of the main results we state criteria for strong (non)oscillation, examine a generalized Euler type equation, and establish criteria of Kneser type. Examples from q-calculus and h-calculus, and a Hardy type inequality are presented as well. Our results unify and extend many existing results from special cases, and are new even in the well-studied discrete case.