"Network" is an insurance company word only. In/out of network is their own network, has nothing directly to do with what is/is not covered or with specific geographic location.

It sounds like your specialist will have to try for another out of network single case agreement (doesn't necessarily mean you can only do it once, could just mean "this" case) if you want to go to that hospital again. But, your insurance may/may not cover it. There's no requirement that they do.