The Stonebow Inn is on the west side of the Brandywine River, so the travelers are already across the Brandywine, on the west side; which means they're in the Shire proper (no longer in Buckland).
According to a detailed map I have, The Water empties into the Brandywine a mere mile north of the Great East Road, and may or may not be fordable, depending upon how creative we want to be. However, ten miles west of the Bridge, is the road to Budgeford, on which one passes over a bridge to cross The Water. Once through Budgeford, it's about a dozen miles to Scary, after which one can take the northeast road for about 2 miles to Quarry, which appears to be a village instead of an actual quarry, best as I can figure. North of Quarry one can cross over fields for about 15 miles and eventually come to Dwaling. North of Dwaling are The North Moors. However, just about 6 miles along the westward road out of Dwaling toward Long Cleeve is a little burg simply called "camp". That might be interesting. So there are some options. Notice that from the Great East Road to The North Moors it's a good 25 miles. That's 2 days' journey, especially for adventuresome and frolicking hobbits. Brown might get frustrated with the slow pace, I might imagine.
