Odd's Ratios in a Hypergeometric Distribution


I have a sample of survey respondents that were chosen without replacement and make up a relatively large proportion of the total population they were drawn from. There are also small expected values in some cells of my contingency tables. Because of this I have been using Fisher's Exact Test to determine the association between different categorical values with nominal and ordinal answer choices.

If I now want to get odd's ratios for these relationships, can I use a normal logit or ordinal regression, or is there another test that somehow adjusts for my hypergeometric sample?

Thanks very much for any guidance you can provide!
Also, is it somehow possible that the fact that I'm talking about more than a 2X2 contingency table disqualifies odds ratios? I have more than 3 category options for both my predictors and dependent variables. I have used these regressions in the past and SPSS just codes my independent variable categories as dummies (binary), no matter how many category choices there actually were. Does this mean it is automatically solving for this issue? Is it even an issue (I read this online but very likely may have been confused)? Keep in mind I also have low expected values for some cells because there are very few respondents belonging to certain predictor and dependent variable categories.