Well, I'd like to say "go away," but we do need it. Its just this time of year, I suppose.
I find that I enjoy Autumn more and more each year, but I just can't make my peace with the rain. It chills me to the bone.
So what do we wear in the rain? If it's chilly, wool is certainly more than adequate. A heavy cotton if it's not too cold, but moving into October, I'll stick with the wool or wool/silk blend. Stockings, of course; patterned and colored tights are gaining in popularity, but not strictly an "authentic" vintage look. I see many lovely versions for those who want to do "vintage-quirky." To each his own. Also at this time of year, I'm quite fond of layers. A sleeveless top with a cardigan, or short-sleeved silk with a cardigan or pullover-all good options. A scarf is a great way to add some color and it's useful.
Now to the footwear. This is perhaps the stickiest point for me in cold weather. My feet are perpetually cold. No matter the outside temperature and no matter the Season. I have cold feet, and if there is any precipitation, so much the colder. The Gaytees boots were all the rage, and I was lucky enough to score a like-new pair in black, low heel with a "fur" trim. I also have a knee-high pair of Eskiloos, lined in sheepskin. These babies are warm. Both of them. I can with only stocking feet in them, and my feet are toasty, no matter the weather!
(this is a 1930s ad, mine are 1940s-50s, so a bit different)
A good trench coat is a must. London Fog, Burberry, whatever your checkbook allows! Traditional tan, with a zip-in fur lining, and belted. It goes with everything.
The hat is optional, I rarely wear one in the rain. My hairstyle just doesn't support it very well, and in the nasty weather, why take extra chances??
Besides, I generally get "door-to-door" service when it's foul weather...and so should You. Insist upon it.