Here's an alternative route I'd suggest from Sway to Damerham, all on road, avoiding the A31 and Ringwood. Nice section through Poulner and the Gorleys up to Fordingbridge.
Also you don't really want to ride that section of the A350 from just after Hindon to Crockerton - lots of heavy traffic.
Turn left at your mile 43 marker and re route via Monkton Deverill, or go via East Knoyle, Mere and Maiden Bradley
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www.plotaroute.com/route/268523
Turn left at your mile 43 marker and re route via Monkton Deverill, or go via East Knoyle, Mere and Maiden Bradley