This is crazy, but...
Sometimes when chatting up a stranger, there's an unspoken understanding that you're not "green," that is, this is not your first attempt (or your fiftieth attempt) at getting a woman's phone number. Some men will use the approach "This is crazy, but...I saw you standing there and I just had to come talk to you." Though this is a misuse of the word "crazy," and really a throwaway line, but it does imply you noticed something special about the woman, and it shows a little bit of vulnerability. Perhaps another case of what you say being less important than how you say it.
(Fortune favors the bold.)
Being bold may improve your chances of meeting someone you connect with.
I'll refer to the men at Simple Pickup, a rather extreme example of being bold. One of the key elements to their success is simply that they don't give a shit what people think of them. While they can be rude, dismissive, contrarian or just plain baffling to some of the women they meet, they're never hostile. However, they are bold and a somewhat confrontational. Now, I've reached the point where I'm comfortable talking to most people in most situations, and I don't tremble in fear when a woman I'm attracted to walks by. I've run events and have a few small public speaking engagements each year. That said, the idea of dressing up in a costume and hitting on every woman passing by the boardwalk, well, that makes me very nervous, and it probably makes you nervous too.
Let's examine why. For most of us, there is a reasonable fear of rejection, of embarrassment, of looking like an idiot in front of women, our peers, or whoever witnesses our attempts. We each have specific things at which we'd rather not perform poorly in public: giving speeches, dancing, singing, playing sports. However, there is a special fear of failing with women in public, as it relates back to our sexuality and perceived desirability. Somewhere in our reptile brain, our inner Beavis and Butthead voice snickers at our attempts: "Huh-huh. You're never gonna get laid. You suck." This may be our internal dialog, but we're also worried that others are thinking the same thing, and seek to avoid the disapproval of both men and women.
Overcoming over this fear, and learning to be bold, you can improve your chances of meeting someone you connect with.
I'm not suggesting you emulate the Simple Pickup approach. Even if you're naturally extroverted, it may not suit your personality or temperament. However, if you keep the spirit of this approach but not the practice, you'll find a lot more opportunity. In PUA lingo there is the "five second rule," which states that once you make eye contact with a woman, you have five seconds to walk right over to her and start talking. This is false. If you walk in to a restaurant and catch the eye of a woman who's clearly deep in conversation with someone, you don't have to intrude or lose her forever. Wait until a good moment to approach, but do approach. The point is to talk to her--it doesn't have to be the instant you see her, but do it soon.
Generally speaking, it's good to follow the old adage "'Tis better to try and fail than not try at all." If someone catches your eye, and you're not at a funeral or dodging enemy fire, go for it. Just walk right up and introduce yourself. You will fail some of the time, that's a given. If you never go for it, you will fail 100% of the time.
For a look at a different set of pros (actors and musicians), watch the crew from CDZA being bold, musically.