The New Forest

"[...] one of the largest remaining tracts of unenclosed pasture land, heathland and forest in Southern England, covering southwest Hampshire and southeast Wiltshire. It was proclaimed a royal forest by William the Conqueror, featuring in the Domesday Book.

Pre-existing rights of common pasture are still recognised today, being enforced by official verderers and agisters. In the 18th century, the New Forest became a source of timber for the Royal Navy. It remains a habitat for many rare birds and mammals.

It is a 28,924.5-hectare (71,474-acre) biological and geological Site of Special Scientific Interest. Several areas are Geological Conservation Review and Nature Conservation Review sites. It is a Special Area of Conservation, a Ramsar site and a Special Protection Area. Copythorne Common is managed by the Hampshire and Isle of Wight Wildlife Trust, Kingston Great Common is a national nature reserve and New Forest Northern Commons is managed by the National Trust."

Cit. en.wikipedia.org/wiki/New_Forest

Since it's creation by William the Conqueror in 1079 the New Forest has been a protected area. It became a National Park in 2005.
Watch out for the wild ponies and pigs which can be on the roads.

Sorry I really don't like the Rodger Penny Way, as a local I try not to ride it anymore..
I'd go through Woodgreen, Lover, Passed Hampworth Golf Course, No Mans Land... much quieter roads... On the way back most of my routes go into Ringwood... and through the forest from there (I live near Lyndhurst)