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The correct answer is the integument system because this system encompasses the skin and its appendages, including hair follicles. The integument system is responsible for protecting the body and regulating various functions, such as temperature and sensation.

One of the key features of this system is its ability to respond to environmental stimuli. The hair on an animal’s body can stand up (piloerection) or lay flat in response to different factors like temperature changes, emotional responses (like fear or aggression), or external threats. This ability to adjust the positioning of hairs is primarily regulated by the tiny muscles called arrector pili, which are attached to hair follicles. When these muscles contract, they cause the hair to stand upright, which can help in insulation by trapping air or making the animal appear larger to predators or during aggressive displays.

Other systems mentioned do not primarily deal with the adjustment of body hair; for example, the nervous system does involve sensory processing and control of muscle contractions, but it does not directly manage hair positioning. The reproductive system focuses on reproduction, and the digestive system deals with nutrient processing and absorption. Hence, the integument system is the most appropriate choice for adjusting the position of hairs on an animal's body.