[souffle]

This is our way and a suggestion to try. If you are managing with the cup to get her out then that is fine but once out you still need to lift and lay. Until they are confident with the lift and lay we don't do the hand to hand work at all. Some people like to use a dep play box and do the lift and lay in that as well as lots of stroking and letting them see that hands are fine to have around.
It's entirely up to you if you want to wake them or not. We do wake ours and none of them seem stressed or have passed away as a result. Bean the hamster in the pics is now fine being wakened and comes out his nest. He has gained nearly 20g since he came and a stressed hamster would not be gaining a good amount of weight. When his family come for him we will teach them how to wake and handle him.
Everyone makes their own choices and using tips from multiple sources and selecting the ones that work for you should give you a good result.
Good luck.