Turnip Boy Robs a Bank

There are 42 total hats in the game and the list with how to acquire can be seen below.

Be sure to fully watch the credits and don't close the game after the final boss fight to ensure you get the endgame hats in your inbox. Hats noted below as only inbox are given to you via the inbox on your phone after completing certain tasks. Be sure to check the tasks guide to ensure you have completed them all.

NOTE: Getting some hats are very buggy. Some have suggested NOT opening your inbox and accepting the hats until the end of the game and after completing all tasks. However this did not work for me, I was still short 3 hats that you are automatically supposed to get in the inbox after completing certain tasks. Therefore I recommend checking your inbox as much as possible. I believe the only way currently around this if you're bugged (at time of writing) is to start a new save. Good luck!

Bald: Automatically start with
Mini Me: Automatically start with
Explorer: (Annie) Finish Say Cheese! Quest - Base Hideout
Turnip Girl’s Bow: (Elderberry Medium) Lead Sister Ghost to Elderberry Medium - Cryptic Crypt
Goblin Ears: (Gregg) Finish Gregg’s Waifu Quest - Dark Dock
Fancy Glasses: (Old Lady Lime) Finish Homewrecker Quest - Elevator
Candied Turnip: Defeat Candy Apple - Failed Festival
Artist Hat: (Starfruit) Finish A Painter’s Payment Quest - Fancy Foyer
Graffiti Hat: (Peas) Finish Tagging Fancy Foyer Quest - Fancy Foyer
Robo-Roids Merch: (Store) Buy from Robo-Roids Store - Hideout
Fox Hat: (Dr. Dogna) Defeat Super Fox - Lab Lab Elevator
Rubber Duck: (Conductor Onion) Finish A Little Sippy Treat Quest - Locker Room Elevator
Coffee Order: (Uncle Rigsby) Defeat Uncle Rigsby - Pitiful Pit
Acorn Hat: (Acorn) During Estate Extortion Quest - Seed Stock
Cryptid Hunter Extraordinaire: (Cryptid Hunters) Give Cryptid Hunters Camera - Seed Stock
Rat: (Rat Friend) Finish Rat Friend’s Quest - Sewers
Golden *Poop*: (Pie) Finish Rocks Rocks and More Rocks Quest - Underground Elevator
Electrician’s Hat: (Watt Melon) Finish Rare Artifact Quest - Vault Elevator
10 Gallon Hat: Defeat Sargeant Yeehaw (Cowboy Courtroom Quest) - Inbox
A Frog: Inbox
A Rabid Dog: Inbox
Alien Glasses: Finish Game - Inbox
Bone: Inbox
Cat ears: Inbox
Chad Sunglasses: Defeat Robo Chad (Longevity Labs Boss) - Inbox
Clown: Finish Game - Inbox
Dillitini’s Fedora: Finish Game - Inbox
Don’s Fedora: Reach Motherlode Vault - Inbox
Flower Crown: (Strawberry) Finish Put a Ring on It Quest - Inbox
Hard Hat: Inbox
Legally Distinct Space Visor: Finish Game - Inbox
Mailbox: (Turnip Boy’s Mailbox) Talk to Mailbox in Cryptic Crypt - Inbox
New Year’s: Inbox
Operator: Inbox
Santa Hat: Inbox
Subscriber Crown: Inbox
Sunhat: (Florist) Finish Put a Ring on It Quest - Inbox
Enoki: (Enoki Mushroom) Finish Mush Cult Request - Inbox
Verdegris Agaric: (Green Agaric Mushroom) Finish Mush Cult Request - Inbox
Fly Agaric: (Red Agaric Mushroom) Finish Mush Cult Request - Inbox
Violet Webcap: (Purple Mushroom) Finish Mush Cult Request - Inbox
You’re on Fire!: Inbox

There are a lot of hats and headwear to acquire. Some are obtained through your inbox so make sure to take care of that on your phone after beating the game. Most are acquired through completing the tasks. Here's a list of the trickier ones with a complete scroll through of all the hats at the end to see what you're missing.

BIG TIP: For those having trouble with the hat achievement, if you decide to do a new playthrough or for those reading this that haven't started yet. DO NOT open your inbox at all until everything else is complete. All your mail with the hats will be there at the very end, I've had a few people confirm this worked for them on their 2nd playthrough and as shown in the video that is how I unlocked mine on both Steam & Xbox.

Also it's being noted in the comments by a few players that if you skip the credits you won't receive the rewards for beating the game so make sure to let them roll.


Hats
Robo Rafael Shop – Robo Roids Merch
Mushroom Hats – Complete Kill & Brutalize Tasks for the Shroominati Shadows NPCs
5:12 – Graffiti Hat
6:45 – Electrician’s Hat
7:00 – Artist Hat
11:18 – Acorn Cap
24:50 – Goblin Ears
29:08 – Fancy Glasses
29:30 – Fox Hat
33:05 - Mailbox Hat (check inbox after)
39:54 – Rubber Duck
41:05 – Turnip Girl’s Bow
46:00 – Rat Hat
47:40 - Don's Fedora
51:50 – Explorer’s Hat (give Annie all Photos)
53:08 – Golden Sh!% (return all rocks to Pie)
53:57 – Accept all hats from your inbox
55:45 – Scroll through complete hat list
Dillitini's Fedora - Beat the game (check inbox after)
Alien Glasses - Beat the game (check inbox after)
Legally Distinct Space Visor - Start final boss sequence (check inbox after)

Photos
5:22 – Fancy Foyer Photo
17:50 – Failed Festival Photo
19:50 – Crangelica Photo
20:10 – Cryptid Photo
21:15 – Dark Dock Photo 1
21:35 – Dark Dock Photo 2
31:36 – Cryptic Crypt Photo
33:50 – Cryptic Crypt Photo 2
38:00 – Soupy Sewers Photo
43:50 – Put a Ring on it Photo
47:40 – Mysterious Motherlode Photo

NOTE: This appears to be bugged for some where certain hats don't unlock upon meeting the necessary requirements. Wish I could help but your best option is probably to head to the Snoozy Kazoo Discord and check there or post in their support thread.

Believe it or not, this is the last achievement that will unlock for this game, if you've completed all the tasks, acquired all the photos and pet rocks, upgraded all weapons and Robo-Roids, and steal at least $1,000,000 from your runs before beating the final boss. Let the credits play through in its entirety, then you'll be back in the Weapons Workbench area. Press to bring out your phone and select your inbox where you can go through all your hats you acquired, there's a few of them that won't show on your inbox until you beat the game, one of them being Dillitini's Fedora. I say this because I made what seemed like a mistake of opening the mail inbox before completing the game in terms of possibly bugging up the achievement, but the achievement did pop-up on screen once I finished the game and went through the mail inbox afterward to find the new hats.

So Nico8912's suggestion to not open the inbox until finishing the game is a pretty good one, the kicker here is it's not possible to get all the hats before beating the game, so just complete all the other collectibles + missable achievements beforehand.