Noisy Transformer solutions
Rewinding is not indicated; the noise is most likely coming from the core, not the windings.

There are a couple of things you can do to mitigate this, especially if the core laminations are visibly loose. One quick thing to try if the core has through-holes is to put long screws through the holes and tighten them down.

Another thing to try is to disassemble the core from the windings and soak it in lacquer to keep the laminations from vibrating against each other. After you soak it, squeeze any excess out in a vise as it dries.

One more thing to try (if you have the resources) is to clamp the core in a vise and run a bead of arc-welding across the laminations on one or more sides. You'll sometimes see this in transformers as they come from the factory.

If it turns out that the problem IS in the windings, you could try soaking them in lacquer, too.

One other thing to try: If there's room, drive thin wooden tapered shims between the core and the winding assembly. This will both compress the core laminations and stretch the windings, hopefully reducing vibration in both.