I make multiple SD pheasant hunting trips a year, a long with lots of local MN hunting. Definitely prepare for all seasons, but being late October you will probably see temps somewhere between 30-60. 60 and sun is too warm. 30 and wind and rain could be cold. 45 and sunny with a slight breeze is really nice when walking all day.
I would bring a few pairs of boots. Muck or rubber boots, and plan on wearing them the most. A pair of waterproof upland/hunting/hiking boots, and only wear those when you are in an area that is higher elevation grassland and you are 100% sure you won’t be coming into water. A boot dryer is certainly nice to have to get the sweat out of your boot before the next days hunt. Moisture wicking socks. I wouldn’t worry too much about the warmness of the boot. Unless it is below 0, you will warm up very quickly from the amount of walking you will do. You can always add a layer of socks if need be, but wet/sweaty feet is not fun.
I think a pair or two of upland pants is very important, however you can certainly make use of your old pair of blue jeans. Upland pants just do a tremendous job of blocking all the brush, wicking away the early morning dew, stopping burs, etc.
Vest – If you are just starting out, any vest with a few pockets for shells and an area to hold birds will do. If you get into the sport, then certainly upgrade if you’d like.
Other: Hat, stocking cap, lightweight and heavyweight gloves, sunglasses, shooting glasses, water bottle, pocket knife.
Gun & Shells: When traveling that far, I always like having a backup gun with me just in case. Know the rules on shells – if you are public and needing steel shot.
Layers: A lot of people make the mistake of dressing for the 30 degrees it is at 10am and not the 50 degrees it is at noon after walking. I usually am just fine in a hooded sweatshirt and t-shirt almost no matter the weather. I don’t like bulk, so I may add a baselayer underneath that if needed, or a lightweight windbreaker type jacket over it.
Depending on the type of trip you are on (remote backroads public land hunting) I always carry a shovel with me. You never know when you’ll get stuck in snow or mud, and how long it may take someone to get to you. Leave each day with a full tank of gas. More water than you think you’ll need. Snacks. Ice. Coolers. First Aid kid for humans and dogs. Zip lock bags for birds. Knives.