How to prepare the geometry to avoid light leaking on large pieces?

I’m a bit confused. If I apply different textures to the same mesh I’m going to get the same resolution that is the resolution of the UVmap / lightmap, right?

Let’s say that I want to apply the same texture to the whole piece, but the resolution is not good enough. If I just apply the same texture to two parts of the same mesh I would get the same resolution. I will need to have UV islands with more density to get a better resolution.