Imho the best time to drive around the New Forest is either c. 5am in the Summer, or in Winter when it’s really, really frosty. It’s an entirely different place.
My favourite 2 N.F. places are the Rhienfield Ornamental Drive, and the Sway Tower.
When you get to Lindhurst, follow the A35 toward Christchurch but after 2 – miles, turn left and head down Rhienfield Ornamental Drive to Brockenhurst. Very pretty in late Spring.
https://www.new-forest-national-park.com/rhinefield-ornamental-drive.html
Once you reach Brockenhurst, head for Sway and, to the south of the village down Barstow’s Lane, take a gander at the Sway Tower – the tallest non-reinforced cement tower in the World, built 1889 – 1885. It’s unexpected – you’re driving down an unassuming little country lane, minding your own beeswax, and then suddenly there’s this bloomin’ great tower! Just – there. In the arseend of nowhere. I love it.
https://www.lymington.com/lymington-life/locally/894-sway-tower