QECP and Whiteways are the closest “good” riding.
You can do some stuff directly from Fareham though, getting up into Botley Woods and just exploring and/or heading down to the coast around (say) Stubbington/Hill Head (parts are a bit stoney by the sea). You could do a loop Botley Woods/cross river at Botley then down to Manor Farm/Park and along the river Hamble and if you’ve time past Warsash and along to coast or just come back on roads to Fareham (A27 is a bit busy). Have a look at the map, footpaths are ok (cough)
You might like to post the same question on this thread too Hampshire Riding