Each employer administers the POSS themselves. Utilities may share the results with other locations, particularly in the case of fleets (Exelon, Entergy). In most cases, however, you would have to take the POSS at whichever utility you are applying at. I've not heard of anyone taking it under the scenario you described. If someone has I'm sure they'll weigh in.
Best of luck with your life after your service. Thanks for your service to our country, & Welcome to Nukeworker!
Tom