Hi Rachel. GEE is one way to account for confounders, but of course you could also do it in a conventional regression analysis. GEE can help with multilevel effects, as an alternative to mixed effects. Effect modification is another possible way to consider SES in a model.
Controlling for clusters within provider and/or hospital is a common use of multilevel analyses such as GEE or mixed effects, while simultaneously controlling for the individual level variables you mention.