Historically, the spring has been a better period to open in if you're chasing Best Musical due to the voters' perceived "recency bias." However, the majority of best Musical winners this decade have opened outside of April.
Last year, the Best Musical lineup was already decided before April got here. Maybe Happy Ending and Death Becomes Her opened in the fall and solidified their status as contenders. They didn't have as much competition opening around them, and voters were able to sit with their viewing experiences of these musicals for many months. Buena Vista and Mincemeat opened in March, later in the season, but still avoiding the mad rush at the end of April. All four shows are wildly different in style and themes. That left one slot open and it went to Dead Outlaw, because voters were likely already familiar with it from downtown, and due to the pedigree of the writers.
Boop, Smash, Old Friends, Just in Time, and Real Women Have Curves all tried to come in right at the end of the season and score that final Best Musical slot. All of them were screwed from the start. And the only one of those shows still kicking is the one led by a major Broadway name. Opening at the end of the season is no longer a way to guarantee a Best Musical nomination. How was Real Women supposed to build up consensus? Nominators saw it once, in the very busy last week of eligibility, and then had to vote a few days later. It's a tall order for any musical to come in that late and say "this is one of the 5 best of the season" when nominators have been secure in their thoughts about a bunch of other quality musicals for months.
I'm very excited for Midnight in the Garden of Good and Evil. But it is a very tough sell for commercial Broadway. And I maintain that opening in the fall, with a really captivating marketing strategy, would be their best bet at both a Best Musical nomination and long-term survival. This is a show that needs breathing room to establish itself. If they open at the end of April, it will absolutely be gone by July.