The only other inn I know of is the Green Dragon:
"The conversation in The Green Dragon at Bywater, one evening in the spring of Frodo's fiftieth year, showed that even in the comfortable heart of the Shire rumours had been heard, though most hobbits still laughed at them." FotR, Chapter 2, The Shadow of the Past, p. 53.
Are there any others, or am I being completely daft?