How to identify polygons with a specific relationship to another polygon layer?

I'm in the process of running a query and ran into a snafu that I'm hoping one of you have encountered before and know how to resolve?

I'm attempting to identify holes in my organization's chapter boundaries. By holes, I mean the zip code polygons denoted by the green X's, not those by the red X's. The zip codes in the green X's are zip codes that are completely surrounded by our existing chapter boundaries. While the zip codes in the red Xs are just zip codes that our organization does not service. I am specifically interested in identify the green X zip codes.

I have attempted to run several different spatial join relationships and analysis tools however, I'm having trouble finding any tool that can distinguish between the zip codes such as those in the green Xs as opposed to the red Xs.