Hair is not an organ.
Hair is a non living keratinous filament which grows from the skin and is not an organ but instead is an accessory structure of your integumentary system, that includes your skin, glands and nails.
Your integumentary system is an organ system, but the hair itself is not an organ.
Hair follicles though are considered an organ and is a complex structure that is located in your skin which produces hair and nourishes hair.
An organ is classified as a structure that is made up of two or more different tissues which work in a coordinated way to perform a function.
Hair follicles are, indeed, made of up multiple tissues and thus qualify as organs but like mentioned above the hair itself is not an organ.
Hair and nails are actually non living structures of the body even though they grow.
Hair and nails are made of keratin which is a protein which is produced by living cells in hair follicles and nail beds.
And once the hair and nails grow out of the skin, they are then no longer living tissues.
While the hair shaft itself is dead, the living hair follicles within the dermis produce new hair in cycles of growth, rest, and death.
Hair is often considered a body part, and most specifically a body part of the integumentary system, that includes glands, nails and skin.