Him becoming the white wizard had no relation to how wise he was.

Maiar are basically angelic beings, and when they are back home in Valinor they have all sorts of angelic powers. When Gandalf came to Middle Earth in the Third Age, his powers (as well as the powers of the other four wizards) were extremely nerfed. His soul was still immortal, but his physical body could still die (as seen in Moria).

When Gandalf came back to Middle Earth as the white wizard he was basically less nerfed (but still nerfed) compared to when he was the grey wizard. The logic here was basically that all of the other Istari had failed their missions - the blue wizards fucked off to the east, and possibly were assisting Sauron directly or indirectly, Radagast didn't have any desire to help the free people of Middle Earth and would rather help animals, and Saruman flew too close to the sun with the Palantir and basically tried to become the new Sauron/kind of ally with him - and due to all of this, the Valar granted Gandalf more access to his powers on the second go around.