Not sure if you should request to remain in the pattern or depart to the north/south/east/west?
If you would like to take off and remain in the traffic pattern, that is what’s known as pattern work. That would be the only time when it would be appropriate to request departure “remaining in the pattern”. here
If you do not want to remain in the pattern, then you would request departure and choose the direction you’d be departing Tower’s airspace.
Tower’s airspace is defined by it’s vertical jurisdiction, lateral boundaries and frequency range for the facility being operated. So vertical jurisdiction is SFC – 5000ft AAL and lateral boundary is the most immediate ring/boundary surrounding the airport.
So when you request departure you would determine which direction you’d be departing Tower’s airspace north, south, east or west.