Do you and the cat slow blink at each other?
To assure her that you are friendly and mean no aggression.
If she returns slow blink, that would mean she trusts your
presence.
Yes, this is important. If you get eye contact with the cat, close your eyes for a few moments and avoid staring at the cat. Closing your eyes signals to the cat that you are not aggressive. This has worked on every cat I have met, it can make a big difference. Closing your eyes, then maybe just look at the cat quickly and close your eyes again. For cats it's a clear and strong signal.
I have have had a few cats that started out staying away from me. And the only things I did was to not try to grab the cat. And giving the cat space (if it's sitting right where you are going, walk around it instead of getting it to move). And talking to the cat with a low, soft voice. Over time the cats I have had and have become very friendly and trusts me 100%.
I think another big thing is food. When the cats sees where the food comes from, if it connects you with food, that also helps I think. So in the beginning when I have had new cats, I have filled up the food bowls when the cat is in the kitchen and can see what I'm doing, So it sees who brings the food. And then I have left the kitchen and let them eat in peace.
You mentioned that she will let you pet her sometimes and allow you to pick her up. That's a good sign. I had a new cat a while ago that was curious about me but became very uncertain when I picked him up. So the trick is to first of all pick them up gently and a little slowly. And one important thing is to immediately put them down on the floor if they begin to twist and struggle like cats do when they want to get down. If she's calm just hold her and the moment you notice she starts to squirm, put her down on the floor quick but also gently. And just repeat that.
I think it's important to never try to hold on to the cats if they start to squirm, they quickly get a little panicky then and they remember it. Maybe they get trust issues. So the next time you try pick them up, they remember it was unpleasant the last time. I think it sounds like you are on the right track, she's alllowing you to pet her and pick her up, it just sounds like she doesn't trust you fully yet. She's being careful and taking her time, not jumping into anything head first. I have also met cats that behaved like we were old friends right from the start, so they can be very different. But most seem to be a little careful, cats can spend a long time checking people out before they begin to trust people fully. You have to prove yourself to them, that they can trust you.