Why this is designated as a Second Edition:
Tolkien revised the text of the Hobbit to match the events in the Lord of the Rings. This newly revised text was printed starting with the 5th Impression.
Why this is considered the second state of the second edition:
The 20th thru 23rd impressions are considered a 3rd state here because a reader in 1964 or 1965 translated the runes on a 2nd edition dustcover and saw that they read “George Allan & Unwin” as the publisher instead of “Houghton Mifflin”. According to Hammond and Anderson this was corrected as of the 20th impression and thus as of this printing it is considered a new state.
- JRR Tolkien – A Description Bibliography by Hammond and Anderson