I think the "next step" is simply to point out in the article that large sample size may be artificially inflating your statistical results. There is no agreement on what a large sample is so there is no test of that nor is there (as far as I know) a way to show what the p value would be if you had a smaller sample size. You could of course take a sample of your sample and show the change in p value associated with this, but I have never seen this done and I doubt it would be a valid approach.

Going on to talk about the effect size, assuming you have a good basis to determine what a reasonable effect size is, would be the logical place to go. Strangely I have never seen that done in the social science literature - but it should be done.