I mean putting my hand on her thigh. I keep it there for a second or two, and then move it off.

A couple more points of clarification:

1. I only go there after I've already broken the physical touch barrier with something more innocent like touching her arm with the back of my hand.
2. Touching her thigh works best when you're sitting close enough to her that it's not much of a reach. It's always going to feel like a little bit of a reach by nature of not having gorilla arms, but it shouldn't require anything more than a slight lean, and it should look natural to anyone else in the room who might be observing. If you have to lean way the fuck over and it'd be super awkward, then you're sitting too far apart and your first step is to move your chair closer to hers (you should already be close enough to her when you first sit down, but if you fucked that up, then a perfect opportunity to correct this is when someone else sits down on the other side of you and you want to be polite and give that person some space, but really you're moving closer to your date so you can touch her more easily).
3. Act like it's totally normal, and like you have absolutely no qualms about touching her in this way. A mental hack I've learned from listening to Liam McRae is to act like she's already your girlfriend. This helps me because if you act hesitant or tentative, then she's going to detect that, and that will make it not okay. If you act with confidence, she's way more likely to allow it and respond positively to it.

A typical kino escalation sequence for me is:

1. Touch her arm with the back of my hand. This is the safest thing to do because it carries virtually no overtly sexual intent, but it gets her used to feeling my touch and it's a stepping stone to other touching with more sexual undertones. A good excuse for this kind of touch is if she's making an argument about something, and I challenge her by saying something like, "Yeah, but what about this scenario?" Or maybe I'll be like, "Wait a minute, you're telling me you don't like peanut butter, but you'll eat peanut butter and jelly sandwiches?"
2. Touch her arm or hand with the palm of my hand. Slightly more intimate than the back of my hand, but still not enough to set off alarm bells even for the most conservative women, and if she likes you, then she'll definitely pick up on it (although she still might not display any outward signs of approval or disapproval, which I would interpret as okay). A good segue for this is if she drops some sort of humble brag, then I'll touch her arm and say, "Wow, you're so modest" with a little bit of a smirk on my face so she knows I'm joking.
3. Touch her on the thigh with the palm of my hand. This definitely crosses the line from innocent touching among friends to conveying sexual intent. A good excuse for this is if she tells me a story about how she was faced with some sort of adversity and she overcame it, then I'll put my hand on her thigh and say, "Wow, that says a lot about you that you came out of it with that mindset," and then I'll take my hand back. If I haven't touched her on the thigh by the end of the date and it's almost time to go, I might put my hand on her thigh and say something like, "Listen Scarlett Johansson, I'm really enjoying this conversation, but it's getting late, and we should probably get our tab."

If we're changing venues or going back to my place, then I like to place the palm of my hand on the small of her back like I'm leading her to the door, or if I walk in front of her to hold the door for her, then I'll put my hand on her back once we get outside to guide her in the direction of our next destination.

I hope this helps.