I am not sure either what you are asking. But let me make two comments.

1. It is not known for which $g,n$ the space $\overline M_{g,n}$ is of general type.

2. It is almost certainly false that if $\overline M_{g,n}$ is of general type, then the same holds for $\overline M_{g',n'}$ such that $g' \geq g$ and $n' \geq n$. (It might seem plausible that Kodaira dimension should be monotone in $g$ and $n$.) However there is to my knowledge no explicit counterexample. But Farkas has proven that $\overline M_{22}$ is of general type and also proved the inequality $\kappa(\overline M_{23}) \geq 2$ which he conjectures is sharp. Also note that Logan's function $f(g)$ such that $\overline M_{g,n}$ is of general type for $n \geq f(g)$ is not monotone in $g$.

So it seems your question is: if we know that $\overline M_{g,n}$ is of general type, is the same true for $\overline M_{g,n'}$ with $n' \geq n$? The answer is yes, and is a special case of the main theorem of: J. Kollár, Subadditivity of the Kodaira dimension : fibers of general type.

As far as I know there are no explicit counterexamples to the stronger statement that if $\overline M_{g,n}$ is of general type, then the same true for $\overline M_{g',n}$ with $g' \geq g$ and $n' \geq n$, but it is almost certainly false: for instance, Farkas has proved that $\overline M_{22}$ is of general type and also that $\kappa(\overline M_{23}) \geq 2$, which he however conjectured to be sharp. Also note that Logan's function $f(g)$ such that $\overline M_{g,n}$ is of general type for $n \geq f(g)$ is not monotone in $g$.