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There probably isn't anything to say after "hi", since you really don't have a valid reason to say "hi" in the first place. You just do so because you want to get laid. And that's the wrong frame!
This is what I would do, assuming:
1. There is nothing situational to comment.
2. I don't really have a reason to walk up to the girl, other than my own sexual urges.
I would walk past the girl, or walk up next to her in the bar, look at her and say "hi" with a smile. The very next second I would ignore her and look straight forward again (in a relaxed and confident way), either keep walking or trying to get the bartender's attention. Let there be a pause, some "space". That makes her more comfortable, and you don't give her that vibe that you want something from her. You are just a cool guy.
If standing in the bar, it's now up to her to show you some interest back. If you have ordered your beer and she still haven't said something, you probably have a cold case. That is when you try to comment something situational or tell an interesting story - and this is the only valid excuse to have canned material IMO. If she still doesn't hook, ignore her.
If walking past the girl, I would not bother about her for a while. I've done my part at the moment, acknowledging her and making her aware of my presence. If I run into her again, I would say something like "oh, it's you again", and follow up with a question like "do you want to grab a beer at the bar?" as if you two are already friends, and take it from there.
I'm sorry dude but that's really lame. Why is going out with the frame of meeting girls to have sex with them 'the wrong frame'? Isn't that what's actually happening or would you pretend otherwise? You are trying to make it look like you accidentally started a conversation with them and that's silly