I do not know if the specific hours are posted anywhere, I did a brief search and could not find them either. I will note that Delta has several flights to JFK, BOS, DTW throughout the morning and early afternoon, a quick search shows Delta flights at 10am, 10:30am, 11am and 12:50pm in addition to your husband's flight. I would be surprised if the counter closes between these flights. That is to say I think your husband will not have to wait for the counter open.