Cove has as many hairstyle options as the Step you're in. So Step 1 has just one hair he can have, Step 2 has two, and Step 3 will have three options. You can see Cove with short hair in Step 3!
https://twitter.com/Patch_Games/status/1296900406417559552
I'm happy you're enjoying the simple slice-of-life experience of the game C: