The thing about rain and being wet is it leads to hypothermia which is insidious and catches one by surprise. Suddenly, you are not just wet but very cold. From workshops on hiking and backpacking conducted by the Appalachian Mountain Club, as well as my own subsequent experience, considerable heat is lost through your head which has a quite good blood supply. Body temperature can be regulated well just by donning or removing a hat. Wet hands also will lose body heat rapidly.

In your case, a shower cap would prevent evaporative heat loss but in a serious downpour water would run down your neck into a rain coat. My own preference is a jacket with a hood that can be raised or lowered as necessary.

A few years ago I was on a 10 day backpacking trip in the Colorado Rockies. This trip was on the Continental Divide at mostly 10-12 thousand feet. One afternoon it began to sprinkle and after a while steady rain. I was moving along generating heat while wearing a rain jacket and pants and carrying a 45 pound pack. I suddenly began to feel cold and within a few minutes, very cold and realized my hands were so numb I worried about getting gloves out of the pack. I had difficulty unbuckling the pack and then unzipping the top compartment where I had waterproof gloves. I did get the gloves on and within a few minutes I was toasty warm.

The point of my tale is to illustrate how quickly the situation can deteriorate. This is why people get in trouble in the mountains. I would look to a decent rain jacket with hood and, maybe kayaking neopreme gloves. On my feet, in cold weather, I use waterproof Sealskin socks with regular cycling shoes. If it is very cold and near freezing, I'm skiing so have not needed or purchased insulated cycling shoes.