Six months does seem a bit too close. One year seems to be a good amount of time for closely timed sequels. LotR that was releasing around the same time made a killing with that schedule. And a few years down the road a more similar situation to The Matrix sequels was with Pirates 2/3. At World's End didn't do as good as Dead Man's Chest but still made plenty of money.
So with hindsight being 20/20 its easy to say a summer 2004 slot would have probably been better for it but only if they had planned that from the beginning. Like others have mentioned in mid-2003 the entire 2004 schedule was essentially locked in place. If you put it in May it would have been demolished by Shrek 2 and faced direct competition with Day After Tomorrow. June it would have cannibalized Prisoner of Askaban for WB's marketing, theater saturation, and audience overlap. July it would have faced Spider-Man 2 and Bourne Supremecy.
I guess maybe they could have moved it into an August slot? Nothing that month would have been big enough to hurt The Matrix at the time.