Agree with Patchwork, we buy enough each year to see us through until late October/ early November. Only once after a particularly wet winter and a series of ponies on box rest did we ever run out at the end of August, we then ensured the hay was soaked before feeding to reduce the nutrient content.
I have fed "this year's hay" in late July, we usually make ours quite early in June.
As this year, I had just about run out of last year's and was going away to a show, so the horse had to eat it, or have nothing.
It was about 6 weeks old, and I introduced it gradually. Normally I would say leave it until the autumn, but he didn't suffer any ill effects.