Any ideas how to indirectly approach girl in office?

[Little Jester]

5 story office building + cafeteria in basement. ~200 people. I work there for over 10 years, about 1/4th of the people know who I am, but I'm only on friendly terms with about half of them.

Cute new girl is on 1st floor (I'm on a different floor). She is most likely an intern who will leave the company in a few months.

I currently don't have projects that would make it logical for me to visit the 1st floor and simply be there to 'spontanously' get in a chat with her. I don't expect her to come up to my floor for something either.

I sort of know two of her co workers on her floor, but I'm not really on friendly terms with them, so I don't really think I can use them to get to her.
She also won't be using the elevator (stairs between basement and first floor is more logical for her), so I won't be meeting her there either.

Only shared eye contact twice in cafeteria in the basement during lunch. Don't remember details about first time, but this time (just now) I caught her looking at me. Our eyes met for a split second and she then broke eye contact and started playing with her hair. I went on to do things as usual (finishing meal, cleaning up) and haven't spoken to her. This is because both times she didn't sit on a table that is 'on-route' when leaving or entering the cafeteria + she has lunch with some of the co workers of her floor (same as I do with my floor), so here it only allows me to go for a direct approach with everyone looking / able to make fun of the act. Not really something I'd prefer in a place that I have to be at every week.

Like everyone in the office is able to, I used an internal db search to learn her name, office number and office e-mail. However, calling her out of the blue on the phone during office hours is awkward. Been there before . Sending her an e-mail feels like a cheap approach, that'll probably not lead anywhere. So I feel like I really must approach her in person first. What you guys think?

Someone have any creative ideas on how to make an indirect approach happen in a location/scenario like this?
I feel like I should take action, otherwise I'm not gonna learn more about her and gonna kick myself later, when she has left the company again...

 

[Tyme2k]

Few things, how often are you approaching girls? How often do you go out and actively get better with women? And how many girls are you seeing right now?

I'all finish my response when I get home.

 

[Little Jester]

Few things, how often are you approaching girls?

I'm not going out of my way to approach. I lately do approach at parties, social venues, especially when I'm alone. I like being out alone without people I know. Actually only then I do a lot of approaching. Otherwise I try to keep things under the radar so to say.

How often do you go out and actively get better with women?

I usually don't actively go out to get better with women. My priories are different. But when I'm out I pursue that goal. I pursue that goal when I'm at my day job, shopping, working out, at a party, whatever..

If i'd estimate, on average I currently get to know 1 new woman each week. The office I'm talking about has this ongoing stream of interns coming and going and sometimes a girl catches my attention. Just a pool I like to fish in and usually I'm able to meet them one way or another. But this girl I simply can't find a way if I stick with the regular options. Like meeting them in the elevator. I want to make it seem low effort to approach them I guess.

And how many girls are you seeing right now?

Don't know what you mean with 'seeing'. If you mean sleeping I'd say 2 girls. One chick I picked up on a speed date, but I'm not all that interested in. And another chick who is chasing me, but she has a boyfriend and I'm looking for a girl I can actually start to build something with.