I'm not familiar with the OMT but there are some nice trails that may be nearer and/or more scenic. I live near Branson and for Ozark Mountain backpacking that's about as close as any for you.
Less than an hour outside of Branson are quite a few miles of trails in the Hercules Glades Wilderness and that's a nice area. And the Buffalo River Trail just south of Branson is very nice. You won't do better than those for backpacking before April.
If I were you I'd look into the Ozark Highlands Trail too. That and the Buffalo River Trail join but it continues on either side of it for quite a bit and through some very scenic areas.
There is also the Sylamore Creek Trail that's nice, and lots of National Forest around it to explore if you want to wander off it a bit, plus, it goes through Blanchard Springs State Park and there's a huge cave there with guided tours that's worth the time, and some really big springs as well.
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