How to get new clothes
The new clothes that can be acquired during The Crown Tundra are spread throughout the set, although much of it comes later. The first ones you'll get are through a few familiar faces. Speaking to Peonia within the Max Lair, she will sometimes give you something that is a sartorial accessory. These include a futuristic eyeglass and some futuristic undertones, so keep talking to him between Dynamax Adventure races and you will eventually get both.
On your first visit to Freezington, as long as you've taken the story to where Peony and you are going on an adventure, the mayor will speak to you as you enter the small town. At that point, he'll give you a boat neck sweatshirt (Freezington's Fame). Now, if you are going to speak with Peony in the base camp house, this will be followed up with him giving you the Expedition Uniform as well. This will be followed by him who will give you a Gold Expedition Uniform once you complete all three Legendary Clues.
The next step will be to wait after completing the Galarian Star Tournament in Wyndon in the post-game for The Crown Tundra. After defeating him for the first time and getting the credits roll, restart the game and find the man on the left in the lobby which can be seen in the image at the top of the guide. It will also require that you have spoken with him in the front station on the Isle of Armor. This means if you don't see him you have to go there and talk to him to get him back here. Now talk to him and he will give you the C x L set which includes a hoodie and patterned tights. If you talk to him again, he will notice that you have caught Regieleki or Regidrago and gives you the legendary RE Hoodie or RD Hoodie respectively.
The most durable clothing accessory to get in The Crown Tundra is definitely the Replica State Crown, given to you by The Crown Tundra Station Scientist. Show them your completed The Crown Tundra Pokédex and they will reward you with the State Crown Replica. This scientist will also give you what is known as the Hex Nut Sports Cap if you have a Meltan or a Melmetal in your party.