I think that the most likely thing going on is that you have a variant of eczema
, called dyshidrotic eczema. Eczema, as you probably know, is an inflammatory condition of the skin that is very common and can present all over the body. Dyshidrotic eczema is a form of eczema that occurs specifically on either the soles of the feet or the palms of the hands. It causes itching and peeling of the skin just as you describe.
Dyshidrotic eczema flares are often provoked by hot, humid weather. It is worsened by the mechanical friction in your shoes that you produce when you go for a walk. Typically however as long as there is not a lot of redness or obvious inflammation it is relatively easy to treat. You will need to focus on keeping your feet clean and as dry as possible at all times. Applying a good thick moisturizing cream several times a day will also help. If there are any isolated itchy spots, you can apply an over the counter steroid cream to those.
If things do not improve with this regimen then you should talk to your primary care doctor
. He or she will be able to rule out other potential causes of peeling (such as fungal infections of the foot) as well as prescribe stronger medicines.