Controlling overall alpha of a material with texture-mapped alpha

I need to control the overall opacity of a game object that has a material which uses the unlit/transparent shader and a texture map to define varying alpha across its surface.

In other words, my object is partially transparent to begin with, but I want to be able to fade it in and out using code.

If it was a uniform color/opacity to begin with, then I could use GetComponent<Renderer>().material.SetColor(), but since it isn't, I'm stuck.

Please help.