The Lord of the Rings is narrated as experienced by the hobbits of the Fellowship. That is why we get a rather detailed description of the Shire. And its inns, of course. There would be the Green Dragon, the Golden Perch, the Ivy Bush, the Bridge Inn, the Floating Log, and All-welcome Inn within the Shire.
Walking out of the Shire, things look less hospitable. There's the Prancing Pony in Bree. The Forsaken Inn is reported east of Bree. Much further south, an Old Guesthouse is mentioned briefly in Minas Tirith, and presumably there are more there.
Are there many more inns? Mentioned, implied, or reasonably to be expected?
After all, folks did travel. Not just the Rangers, but less sturdy individuals, messengers, traders, etc. That alone might keep a travelers' inn or two going even in some of the otherwise uninhabited areas along major roads.
Moreover, I'd assume, in densely populated areas people would tend to gather for an evening occasionally (as hobbits did with some enthusiasm).
On the other hands, hospitality seems to be largely household-based in LOTR: well-to-do houses will frequently receive guests, based on various connections or recommendations. Nevertheless, there are many inns right within the Shire.
I wonder whether one might argue that inns, somewhat like smoking, are rendered primarily as a hobbits' pastime in the books.