Cannot access the 1872 beeerhouse list - so if I am duplicating entries already known my apologies.
On line newspapers are often a good source for pub info. The are several mentions of the "Dropmore Inn, Littleworth Common". Checking old maps I can see a Jolly Woodman, Littleworth Common marked - but not the Dropmore.
The New Inn was in the High Street - "83 yards from the Red Lion". The 1908 newspaper article is about there being too many pubs in the locale and a challenge being made to renew the New Inn's Licence. The landlady in 1899 is a Mrs Alice June Montgomery but the 1901 census entry for her ("a licensed victualler") doesn't identify the house number.