I can only give an answer from a UK point of view, and even then not a very comprehensive one.
At 'my' unit there is another aerodrome within the Class D control zone. The zone is delegated to the Tower up to 1,500', so any departures from the other aerodrome would call the Tower frequency, regardless of whether they're intending to remain within the zone or depart it.
Most of the movements from that aerodrome when there is no ATC present are helicopters, so they have the luxury of being able to lift to establish R/T contact and get a clearance. The gliding club have a procedure which involves ringing up for a clearance and then being airborne in a specified time window. These are procedures for a relatively quiet regional unit though, so a large airport with lots of different frequencies might operate very differently.
It's probably best to ring them up, explain the situation and ask how they usually handle it/which frequency they expect aircraft to call on.