Hi,
There is a record for a William Teague borne in Cranborne, Dorset (not far from Fordingbridge). Born 7 Aug 1778. Father also called William (no mother listed).
Source FindMyPast (Find My Past).
The boundaries between Dorset and Hampshire have moved over the years so looking in Dorset might assist you more. Try also playing with the spelling of the name Teague as there seem to be many varieties out there.
Ancestry show numerous Teagues born in Cranborne about around this date.
Richard