How to change to private just for labour?

my check ups all at KKH, but my labour, i choose to deliver at thospson medical. im 31 weeks preggy. last month i told KKH doc i want to deliver at TMC as i have already booked my doc there. so KKH doc print out all my documents from the start and now for me to hand over to my delivery doc. im officially discharged from KKH.

not sure if u can just change during labour. i understand from my friend they usually switch to private at the last few appt to secure the gynae because u duno when u will go into labour