Cleansers with a chemical exfoliant ingredient can help BUT the acids need to stay on your skin in order to work. Washing your face and then washing the cleanser off doesn’t give them enough time to be effective, so do try and leave the cleanser on for a short while.
Chemical peels use the same ingredients as chemical exfoliators at a much higher concentration. At lower concentrations, AHAs do what you want an exfoliator to do, but they won’t remove entire layers of skin in the way that a peel may. Most consumer chemical exfoliants have between 4% and 15% glycolic acid, whereas a peel is done by a skincare or healthcare professional and uses concentrations between 20% and 70%.
Other chemical exfoliators include:
Retinol (Vitamin A) is sometimes included in exfoliation formulas where the skin can convert Retinol to Retinoic Acid which is a good skin exfoliation and anti-aging agent. Retinol has been shown to improve the visible signs of photo-aging as well as normal chronological aging, when used on a daily basis.
Biological enzymes such as Papain and Bromelain stimulate exfoliation by decomposing proteins into smaller fragments, causing a softening effect to the skin. Unlike AHAs, enzyme activity is not pH dependent but is activated by water and limited in the amount of exfoliation that may be achieved.
Rice Bran (the outer layer on brown rice) is a rich source of biologically active agents that are beneficial to the skin. These include Phytic Acid, Gamma Oryzanol and other important plant actives. Rice Bran may contain up to 10% Phytic Acid which chelates (or binds) calcium ions, helping to loosen cell cohesion and promote exfoliation. Phytic Acid also chelates copper, thereby inhibiting tyrosinase in melanogenesis and so helping to control hyperpigmentation.
Note: I have paraphrased the part of the post above ("other chemical exfoliators") from the above mentioned link.