I have no idea of the origins of the term, but a wideawake hat was a hat with a wide brim that shaded the eyes. They typically had a round-top, rather neatly-fitting crown. Versions of them have been around for centuries.

They were very popular in Australia in the early 19th century because of their ability to reduce glare and therefore fatigue of the eyes. They became the akubra (Australia) and stetson (US) over time when it was realised that a taller crown reduced the amount of heat transferred through the hat to the head.

Sorry I can't give references. I read about this years ago. Somewhere!