The answer is yes, in both dryland and irrigated soils. The root hairs continue to grow at the level(s) where water and oxygen are both available. There is some primary root growth, typically until mid/late August.
Brace roots do most of their growing after pollination. The brace roots often support many fine roots and root hairs IF there is moisture at the 0- to 6-inch level. Most of the active water/N/other nutrient absorption is by fine roots and root hairs that have developed 7-10 days prior.