If I were you I’d go catch the Hythe ferry out of Southampton docks then pick up the NCN2 route (sign posted from the ferry dock off the pier) head into the forest on reasonably quiet roads, keep to the NCN2 into the gravel section and follow through to Brockenhurst, cut behind the Giant bike shop and head out towards Sway, then head north and under the A31 towards Normansland, turn south on the B3079 back towards the A31 then head into Bartley and Woodlands, cross the Lyndhurst road at the Happy Cheese Pub, head south past New Forest Wildlife Park towards Hythe and then catch the ferry back.
That route is about 30 miles, no hills per se’ but it’s bumpy and you’ll break a sweat. You’ll see some lovely areas of the forest, climb on top of the the hill at Normansland then head back on a fast swoopy B road, then quiet lanes through a couple of sleepy villages back into Hythe.
Depending on how you do you might catch the commuters catching the Hythe ferry back to Sotty so bear that in mind, a return trip is £9 I think..
Or if that’s boring come out with me in the South Downs 😀