If you have the Octo Expansion DLC you will be able to unlock the following gear by completing the corresponding task.

Conductor Cap
Get ALL the Mem Cakes in the game.

Golden Toothpick
Beat the secret boss 'Inner Agent 3'.

Neo Octoling Armour
Get ALL the Mem Cakes in 2 Lines.

Neo Octoling Boots
Get ALL the Mem Cakes in 3 Lines.

Null Armour Replica
Get ALL the Mem Cakes in 5 Lines.

Null Boots Replica
Get ALL the Mem Cakes in 6 Lines.

Null Visor Replica
Get ALL the Mem Cakes in 4 Lines.

Octo Shot Replica
Beat the Octo Expansion.

Octoling Shades
Get ALL the Mem Cakes in 1 Line.

Old-Timey Clothes
Get ALL the Mem Cakes in 8 Lines.

Old-Timey Hat
Get ALL the Mem Cakes in 7 Lines.

Old-Timey Shoes
Get ALL the Mem Cakes in 9 Lines.