As an alternative to going west, you could head north east. One of my imperial century routes is from Winchester to Bournemouth and back through the New Forest. It's much more pleasant for the cyclist in winter - in summer the roads can be congested and the locals get all pitchforky. There's plenty of wildlife to see and the scenery is surprisingly varied. If I had 100km to use up, I'd head towards the Bolderwood (deer) and Rhinefield (giant redwoods) ornamental drives.
Downside is that I'e never found a good way of crossing the A338/A35 to reach the Bournemouth/Boscombe/Christchurch coastal sprawl to and from the NF. The A35 is a sugar rush, but I usually have my best TT legs fitted.
So is mine. Avoid the A35 like the plague, and avoid the A31 completely. I did the stretch of the A31 from Ringwood to Burley, for a laugh last week. It wasn't funny. If you want to go up into the Forest, head for Christchurch from Bournemouth, then turn towards New Milton, then head off towards Sway and Brockenhurst, then turn off towards Beaulieu, when you clear Brockenhurst.
https://www.relive.cc/view/838329575
The first bit of the return leg is good, until you get to the A31 between Ringwood and Burley. That's not fun, unless your idea of fun is getting swiped by a Stagecoach double decker bus ( yet again ).
https://www.strava.com/activities/288612641
That's a better way of doing it, it takes in some nice scenery in the forest
