After a girl acts aloof, do you guys actually give second chances?

I'm going to summarize this real quick. If you do not care about the contextual background you can skip it.

Contextual story:

This happened in the gym which I frequent. I approached this really hot babe 2 weeks ago, I reopened her thrice actually the same day, which was caused by boredom from my part. No other hot viable girls around. Anyway she was polite and friendly, but not really contributing. And the last time I opened she acted slighly annoyed. I called it quits right there and then and went ahead and became social with others. I always maintain my vibe and frame, not broken like some chump, for me it was a low effort interaction. I also notice most guys are intimidated by her, not me though. She's not dealing with some average low level kind of guy.
Anyway 5 minutes later I see her talk with this guy friend of her and being warm and really interacting, ofcourse I take note. Later on when she leaves the area she doesn't acknowledge me, I return the favour. No hard feelings but for me that was a dead end.

Fast forward to yesterday. I had already mentally cut her off. So I walk around just doing me, training relatively hard and making fun with general people. I notice her noticing me with a fast stare, I don't really put effort that moment. At one point we were in the same area, but I avoided eyecontact and maintained distance, not going really out of her way, but not putting myself in her proximity either. Pretty well balanced I think. I think she noticed. Meanwhile my vibe was more like "I'd rather not." "Hmm, I've been this route before, didn't seem that worthwhile to me". I left the area and I could feel her power shrinking. Near closing time I was fooling around with the staff of the gym at the counter near the exit and we had our laughs during some juicy gossip. Suddenly the hot babe, walks to the exit and gives me a clear look while doing so, which I can see as a approach invitation. But for my feeling it was a little too controlled, more like "I expected you to talk with me, talk with me", instead of "I really want you to talk with me". There is a difference if you ask me.

Actual Question:

If a girl acted aloof and stuck up the first time you talked with her, do you ever give her a second chance when she throws you signals and if you did how did it work out for you? Remember I reopened her thrice,albeit the same day and she didn't really hook. I have trouble with believing that it is "real". Because if you are not appreciated the first time around, why would you be so the second time? I have dabbled with this in the past and never really had this working out in my favor. Maybe after some long ass months when the girl had tons of time to reevaluate me, but I'm not about to wait on that. I can reengage her, but my feeling is that she is not "weakened" yet. Like throwing a pokéball at a pokemon with half hp full, the pokemon is just going to escape.
I'm not sure what I will do with this girl, she is on my lower priority list despite her hotness, I think I will escalate with compliance requests to filter her out fast. You can't be a chump when you ask compliance even if she refuses and when you back out they'll know why. What I want to prevent is her seeing me as a random source of validation. I'm just really skeptic because of her bad first impression to me.

Thoughts?

you can totally give her a second chance. When I first started and if things didn't go well the first time I'd get a bit of a hurt ego and so I'd be the one auto rejecting. But you know, now that I do the whole second chance thing, it does work a lot of the time. Just be cool about it. Because at the end of the day you're either going to get laid or you aren't and sometimes you have to go through that whole second chance thing. Girls change their minds, a lot, depending on their emotions. So yeah, go for it.

That's what I am worried about, trying again which means that I will invest more. At one point they become more important than they ever should have. I want her to invest in me, not the other way around. I could sense her frustration and I am pretty sure that counts as investment, I'm just not jumping up and down out of joy about getting in an unnecessary tug of war. Because that's what I think it will end up like. Unfortunately girls like these have led me to war of attrition kind of situations where even after you have won, there is no true victory. Just a desolate battlefield with lots of scorched earth.

There is one thing I am sure as hell going to do though and that is make it clear that if I am going to engage her again, I ONLY engage her again because she gave me an approach invitation. Not because I was waiting for a wondrous opportunity which she was so very nice to give me like a philanthropist or something.I actually have some really solid girls at the moment, just like in Varoon's words: I like to focus on the green girls, not the orange ones which make it unnecessarily complicated.

Thanks for your thoughts Jacob,

Anyone else?

Do it with the mind set of being the bigger person and acknowledging her. Just a simple "Hey there Alyssa", as you are moving to somewhere else with purpose. Sometimes it takes them by surprise. Think in terms of how you make them FEEL. Let that guide you.

The drive by "hello" lets her know you know her and acknowledge her but she doesn't possess a power over you. but you made time for her...awwww..

I deal with this with customers who think they are something special. I make it a point to engage them as I'm on my way to something else, and not hover. I have a boss who fawns over this kind of customer and it doesn't work. If they feel you are open to them but not pushy then they seem to be more outgoing...

By all means if she engages you, great build on that.

Hey man, I have been heavily involved in sales myself so I get what you say. I think my reluctance to this girl is because I sense that she has thrown some bait. She puts the ball in my court despite not having really invested herself.

I think your way of calibrating is indeed better than to straight up ignore her. I think I did a great job by not showing her that she had power over me in not a defiant but "whatever" kind of way. I will use your approach, because I indeed sense this broad thinks she had power over me, she tries to steal the frame back. A frame she never had by the way haha.

Still, I don't want to really focus on her or make great mental gymnastics. Her bait just seemed, and probably is, not genuine. So I wanted to hear your views.

I will use your approach Fuck this, but I will make it a little more less effort even. Like give her a slight cocky smile kind of acknowledgement when I walk through. I think saying "hi" is showing too much interest to her at this point. I mean she just gave me a weak ass stare, she can pick her lazy ass up and say "hi" as well. Tons of girls are doing it, so she can as well. I only show more consideration to the nervous girls who are worried about my attainability.

Just goes to show that you should never ignore your instincts. Thanks for the feedback bro