If you’re seeing a texture on the object, then it’s already been UV mapped. What’s happening here is that you’re replacing the existing UV mapping with your own, new UV mapping, which changes how the existing texture gets wrapped across the mesh.
Note that, when you talk about fixing that particular area, it kind of looks like a part of the mesh that doesn’t have good UV coordinates. Like they were seamed verts that lost their seam. If this is the case, texture painting isn’t going to help-- every time you try to paint to fix it, your fix will screw up some other part of the mesh. Instead, you’d need to fix the existing UV mapping.
If you want to keep the old texture, and apply something new on top of it, you might consider that you can make a whole new UV map for the model. Check out the data subsection of the properties view, UV maps subsection. Here you can + make a new uv map, select it for use/unwrapping it, and create an additional UV map for your model. Later on, you can bake your new texture to your original uv map if you want (assuming that old uv map isn’t too messed up.)