Depending on what you want to do, you've got options.
Up at Alton is Brick Kiln Farm which is a pay and play site, which seems to be open pretty often and the last time I went with Hound Dog was £10 a car. There are also sites just over the border into Sussex at Slindon and Boxgrove http://www.4x4driving.co.uk/locations.php
Also if you have a look in the South East section, you'll see the threads for our own woods site where we run events quite regularly and they're always fun. I go up from Southampton for those.
Green lanes, again you need to head North but there's a good selection around the County, have a look at the HCC [url]www3.hants.gov.uk/locating-row/row-online-maps.htm[/url].
Once the weather picks up a little more I'll start getting out and about in the lanes and you're always welcome to come along.
Hope that helps