Logging including session name

[GreudeFumat]

There are several ways to save the nave of a log session, but none that could allow the session name itself.

Describe the solution you'd like
I am heavily dependent on some scripts to run HealthChecks and various other operations via a gateway server to connect to other remote servers, and all session are logged under the gateway server IP. As (my) session names are personalized under specific scripts connecting to specific remote servers, would it be possible to add the session name into the log name output ?
Any feedback would be highly appreciated. Many thanks!