Dog Rash/Thinning Hair

We rescued a small dog, appears to be a chihuahua/terrier mix, weighs approx. 6 lbs. He has scaly/dry skin with thinning hair, but not spots of hair loss. It hasn't gotten any better or worse since we began bathing, feeding, and caring for him. There is a small spot between his eyes that appears to be red and dry, but the rest of the skin seems to have a normal color. Could it be something serious? Any suggestions?
Thank you!! :)

[Aleda M Cheng, D.V.M., C.V.A]

Have a skin scraping for demodex.  These mites are not itchy but can cause thining hair.  Also, have the dog tested for thyroid disease and other endocrine disorders.