We are in Central Indiana and if the weather cooperates I like to bale in Late May. That is when the hay quality is best, but the weather doesn't always cooperate. Most people don't seem to bale until a month later in our area, which means they are baling over-ripe hay that has lost a lot of nutrition.
As for baling up the dead grass, mowing is probably still the better way to go unless it is still standing nice and tall. It will be either a dustfest or you'll end up with moldy bales. In either case you won't have much to work with at the end. If you mulch it up good with a mower the new growth will come right through and the old growth will decompose and feed the soil.