You really don't have
to hit much pavement at all as most of the trails around the entire area are interconnected and there are certainly places to start within a mile or two of pavement to get you to others. There are also trails that start right at Byrds that can lead you to the same places with nothing more than crossing a paved road. It will just depend on who you're following and where they want to go.
The only place you might like to go that would involve any pavement at all would be the Oark Cafe to get some pie, and that's only about 10 miles from Byrds