It's not in the historic district, it's about 2 miles away. Look for places along Bay St or adjacent streets or go across the river to the Beaufort Bread Company on Ladys Island just across the river. While on Bay St, stroll along the riverfront or stop by Old Sheldon church Ruins out near Rt 17 on your way to or from.
There are two in downtown not even a block off Bay Street... Palm & Moon Bagel Company is good, but for a bigger menu, you'll want to try Blackstone's Cafe on Scott Street. After breakfast, be sure to check out the Waterfront Park and grab a coffee or smoothie at Common Ground. I also second the recommendation to see the Old Sheldon Church Ruins as well... they are basically on the way back to the interstate.
If you have time, give yourself a few minutes to drive (or walk) through The Point historic district, which is across Carteret Street from downtown. There are some beautiful homes and gardens to check out.
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