Hanbury Hall, National Trust House & Garden
Hanbury Hall (OS Grid Reference SO944637, see map by Streetmap.co.uk) is even further away from Warwick University than Coughton Court, but is still easily visited by car.
A nice little garden (above), while this side view (below) shows some of the "false windows" which are painted onto the outer wall to maintain the pattern. Internally there is a fireplace instead:
Below we have the orangery - a rather fancy greenhouse which really does support orange trees: