The cage she is in isn`t ideal, no. But it may make things easier in the short term for handling/taming purposes. I wouldn`t put her into a larger open space right now as she is likely to lie flat and freeze. So possibly the best thing would be wait until she is awake, usually later in the day as they do like to sleep during the day, hold something made of fleece, like an old glove or a small folded tea towel and rub your scent onto it. My female loved to walk into an old hamster pocket hammock I had and this was her little `cave` where she felt safe and I could lift her up for a few minutes and then gently put her down again using the pocket hammock. I would do this method again if I had a Chinese hamster.
Once she has a daily routine, you will find she will emerge more and not be so shy. But remember too that it can take a while, so if you need to go back a step, then do so. Don`t feel you need to rush. It`s all about allowing her the time to come to you.
And cage-wise, a Ferplast mini Duna tank would be good right now.
