May be splitting hairs over the definition of the girls portion of the session fee, I took the OP’s question as more on the timing of delivery. I don’t know if any full AMPs that will continue the session without the full amount up front. I hate the buzz kill dealing with it directly brings, which is why I discretely lay it out.
However, I’ve had very disappointing results from attempting the same in a sub-AMP. Even though I’m equally uncomfortable openly discussing services and payment, I haven’t figured out a better way.
As far as $ over an above, I’ll frequently pony up more after the session to recognize a great visit or set the stage for the future. Typically $50 or less, but have gone as much as $100, if the little head is still riding the wave of euphoria.