Should a chi-square goodness-of-fit test be followed up with Odds Ratio? I am wondering what is best practice for reporting a chi-square goodness-of-fit test. Should a result be significant, would a follow up of an Odds Ratio be required if there are more than 2 categories?

It will depend very much on your specific question and dataset. You may want different ORs (which ones to report is a question in itself if you have more than two categories), or it may be useful to show plots, or you may want to report tests on specific regression parameter estimates, or specific linear contrasts. There is no one-size-fits-all answer, you will need to consider your context.
