i hate to be over using this answer, but if you were a plate spinner you wouldn't have this problem.
the reason you have this problem is because you have anxiety over her reaction. you start thinking 'what if i go to hold her hand and she doesn't like that. ill look like a fool!" in essence, you look at yourself a some lowly peasant and her as some magestic princess, and you keep worrying that at some point the gig will be up.
whereas if you were dating other women also, you'd instead be thinking 'what, she doesn't want to hold hands? thats lame. i think sally and jen will get more of my attention moving forward since they enjoy holding hands."
you completely change the frame.
and trust me, most women will do whatever you want them to. UNTIL that is you start to exhibit hesitation, fear, anxiety, etc., then they smell the AFC in the air and they know they own your balls. this is when they start making you look like a fool.
so anyway, long way of saying, JUST DO IT. trust me, if she's gone out with you three times now she will be more than happy to hold your hand (and probably your penis also).