How about this? instead of holding hands, offer her your arm. That way she can if she chooses to instead of being awkward about actual skin to skin contact.

Just ask her while you're walking together. Say this, you know they say chivalry is dead but I don't believe so. May I offer you my arm? then extend it towards her. Now if she takes the arm, you can retract your arm and she'll be even close to you than if you were simply holding hands.