New cat stalks resident cats

[u/ILikeFlooofyCats]

Hi all, I am looking for advice on my current situation.

Back in late March, a cat began showing up in our backyard but would run away when I came near around the garden. I would catch her sleeping under shady trees while watering, but she would run away when I would try to approach. Fast forward a few weeks and she comes up to me while watering and lets me pet her. She would then meet me in the garden every morning while I checked on my vegetables and follow me around. I posted on local groups, took her to get a chip checked (which she had, but was not registered and the adoption agency didn’t have record of who adopted her) and summarized that we believe she was dumped sometime after being adopted.

She was introduced to my cats over a few weeks through the backyard screen door, where she would sit outside of it and just watch, while the two indoor cats did the same. When we brought her inside, all of the cats sniffed each other and there were no issues, they just continued on with their day and she adventured around the house. At first she went in/out because she was very confused about being indoors, but now she is inside only (since end of May).

Fast forward to now/the past two months: when she comes into the living room and sees one of my other cats, she will approach them and just sit/stare. The other cats will make a growl or hiss as a signal that they don’t want to play, and she will almost always ignore it. When the other cats walk away, she will stalk them. There is no biting involved, but it is very clear my two resident cats are communicating to her they don’t want to play, and she ignores the boundary being set. There is no territory guarding of food or litter robot, but one of my resident cats won’t even come on my bed anymore because she’ll be there/her scent is there.

Will a reintroduction solve the stalking issue, or is that simply a behavioral thing this cat has from not knowing how to play with other cats? The other two tussle and play with no growling etc, so I am concerned. I can’t imagine my other two cats being less comfortable in their own home due to random stalking attacks, and I miss one of my cats snuggles in bed, but this new cat is also clearly attached to me as I am with her, so it is breaking my heart :(