I've just had a quick flick through the Tasmanain Trail web site, and of particular note to this forum is the camp sites shown on the map page. You can click on the boxed sections of the map to see a larger version of each section, and from there, you can click on the camp site markers (or at least some of them) to see photos of the campsites.
From what I can tell (without having yet looked at a better map), these campsites appear to be near roads and some of them look like good spots for car camping.