After a chemical peel, regardless of whether you're going out or not, you must apply a broad-spectrum sunscreen because the damaging UVA rays come right through window glass, and after an effective peel, you've removes everything from the surface of your skin, including dead cells and sunscreen. As a result, you need to apply sunscreen. I also recommend applying a skin-type appropriate moisturizer after the sunscreen.
Superficial chemical peels like glycolic acid peels, combined with daily glycolic exfoliation and daily moisturizers, will optimally restore the water balance on the skin of the back of your hands and relieve dryness. This reason this works is that glycolic exfoliation removes the accumulated dead cells that prevents your moisturizer from working. Wishing your smooth, well hydrated skin!