What additions can I make to my basic white bread recipe to increase its fiber content?
You can buy wheat bran at health stores. Before adding to bread, soak for a while to rehydrate. It will change the character of the bread somewhat, but the recipe should still work with something like 1/3 cup. The bag I bought was fairly coarse, so I put it in the blender and pulsed to powder it.

You can also buy soluble fiber (i.e. benefiber in the USA), although some of what is sold might be a scam. I imagine something around 1/4 cup added wouldn't mess up the recipe.

King Arthur (and perhaps others) make a white wheat flour, made from an albino wheat with husk that is ground very finely, and can generally be subbed in for white flour yielding higher fiber product. You might want to start half and half white wheat and white.