It should not be necessary for the Dispatcher to 'Sing Out' from Service Board when new Dispatcher Access has been configured on a Service Team in Salesforce in order to have access to the new Team and Technicians.
Now you must Sign Out and then start Service Board again. A F5 to refresh should be sufficient.
What is the underlying problem do you intend to solve with this idea?
Avoid frustration with users. Even Dispatch Console will work next time you start it
How is the problem being addressed today, if at all?
Inform users on tickets they create that it is not working.