This is intriguing; I wonder if someone local can check it out. The excellent ‘Hampshire Treasures’ resource gives just the following information – “Monolithic Stone – Moved to roadside during road making operations”.
It’s in South Baddesley, which is part of Boldre parish – the “Bouvre” of the Domesday Book (1086). The parish contains evidence of habitation during the Stone Age – implements; Bronze Age – barrows; and in Roman times – coins found at Norley wood.
The stone was indeed moved to the side of the road and can still be seen.
Baddesley is probably a corruption of "Baeddi's leah" meaning Baeddi's forest clearing, as opposed to North Baddesley which is a corruption of Bede's Leigh, since there is a connection with the famous Saxon monk Bede.