This is a good question, in ecology, for example, one might have a single set of data and they will scrutinize a model for years and find the "best" one. In genomics, we choose one model and apply it to N (million) probes / measurements / genes / CpG's / SNPs etc.
You could look at the distribution of p-values, either of your covariate of interest or of the potential confounder with and without the confounder. You can also look at the distribution of covariate estimates. If you see genomic inflation without correction for some important covariate, then that might be a good thing to check.
You could also do something like SVA or PEER on a simple model and see how the inferred surrogate variables correlate with the covariates that you did not include in the model (this is similar to your PCA idea).
Of course, it could be that different covariates only have an effect at certain sites in the genome--we know this will be true in the case of, gender and the sex chromosomes. For other effects, it will be more subtle. In the end, you will have to rely on biological insight to a large extent.