Three days is still very early
Lots of good advice above. We did bathtub taming until our Syrian got used to being handled. If they are scared they may bite (or threaten to) and also if they think you are food! I think it sounds like the latter with your syrian by the way you describe her holding the finger in her paws first
In the bathtub they can run around safely without escaping and you can occasionally stroke her on the back and see if she tolerates it. Then move on to cupping a hand under her belly but just letting her walk off again. Once she tolerates that you should be able to pick her up.
The other thing they can do before fully hand tame or before they are used to you, is "ping" which is wriggle and leap in the air so you can't keep hold. They can ping quite high so make sure any handling is done in a place a) she can't escape from if she leaps and b) that she has a soft landing and isn't being held too high up. Eg over towels in the bottom of the bath or over cushions or the sofa. I used to block of the ends of the sofa with a big blanket so ours couldn't leap off.
Having a hamster ball handy is also useful to put in front of them if they look like they're doing a runner. And as Amethyst Ice says, for getting them in and out of the cage or for carrying them in to the bathroom for bathtub taming.
They often don't like a hand in the cage and may run at it before tame, but are fine with hands out of the cage. I also use a cardboard tube for getting a hamster out. Putting a treat at the far end and letting them walk into the tube then put a hand over either end of the tube and then either lifting them nearby to a playpen area or the sofa and putting the tube down there. Or putting one end of the tube inside the hamster ball and lettig them walk into the ball that way, and then pop the lid on and take the ball where you want to let them out.
An old pringles tube is ideal.