Well first of all those dark spots didn't pop up over night, it can take several skin cycles before the damaged tissue is sloughed off , and the dark spots start to fade from the chemical peels; the key is patience and using effective ingredients consistently. An effective home care routine should include ingredients to help lighten the existing dark spots and also ingredients to help inhibit the overactive melanin [ pigment cells] that are causing the dark spots. Your skin professional should be able to advise you on these plus a series of chemical peels spaced about 3-4 weeks apart should really make a difference.
Peels can be used to improve the appearance of acne scars, melasma, sun-damaged skin, wrinkles, and several other conditions. They can also be used to lighten dark spots.