When the result is "non significant" you don't say "how much insignificant it is" besides quoting the p-value.
I think, that what you want to do, is to informally test another hypothesis, that the parameter is significantly smaller than some chosen by you threshold. In that case you'd need to formally create another test for it (which in most cases would should be straightforward) and quote this new p-value (which we expect will yield significant result).
The bottom line is, that the classical statistics (contrary to common sense and every-day intuition) doesn't provide arguments for null hypothesis. It can only give arguments against it.
Take a look at Bayesian Statistics. Although much more difficult to implement, in the end it does provide arguments both for or against any tested set of hypotheses.