A bit more detailed:

It does not matter if you select vertices or UV vertices. So the selection can also be done in "Modeling" mode. And the selection does not have to be very precise. It is only important that all required vertices are covered. Nevertheless, one should select as precisely as possible. The best way to do this is to have vertex groups in the model that you can select.

After creating the new material (which is then empty), select the main material that covers the imagemap in question (preferably in "Shading" mode). Copy this material into the newly created material.

Back to "Texture Paint". Now make sure that the correct image is selected. Then you can start painting.

If you have to paint the same figure more than once (i.e. combine several different material zones), it is worth making a copy of the blend file. You will only need it to create the textures. This also has the advantage that you can remove material properties (such as specular) that tend to interfere when painting textures.

Oh yes: Save more often is the order of the day. It's annoying when Blender hangs after hours of painting or something else ruins all the work.

---