National Highways says the bat tunnel will help wild animals "safely navigate" a dual carriageway.
The A34 near Winchester remains closed southbound between the A303 and the M3.
Motorists had faced delays heading from Kent to Essex, National Highways says.
Despite never being used by a single vehicle, defects such as surface cracking have already been found.
National Highways says the improvements will improve safety and journeys for road users.
The A31 was closed for two weeks following the devastating wildfire in the New Forest in August.
Work will begin on replacing the M6 Thelwall Viaduct's joints on 1 September.
Knights Bridge is named after Valerie Knights, who came to the aid of accident victims on the A47.
Several organisations reported losses after the A31 was closed due to a wildfire in the New Forest.
