As a few posts have said, one very popular way to deal with this is by a luggage rack which has a sidescreen stowage facility.
I didn't want to get one of these because my personal view is that the extra width disturbs the lines of the car, so I have slightly different set up.
I use a fairly standard lowline rack and a shaped, padded waterproof bag. I got mine from Heart of England Morgans but other dealers offer them.
The one 'extra' I did add was a sheet of shaped foam to go between the sidescreens when they are in the bag. The perspex is very soft and this prevents scratching
I find that I can strap the bag onto the luggage rack and still have room for a small suitcase and also the bag which stows my hood/tonneau.
The reason for a smallish case, rather than something like Steve Simmonds 'American Tourister' item is that I don't like a lot of weight overhanging further out than the rear axle /suspension because in my car's case, it alters the steering 'feel'.
The case fits into a shaped, ripstop nylon cover which I had made, complete with cloth Morgan wings badge.
If I want extra luggage storage, I have a set of Rob Hills' shaped bags which go behind the seats
I've used this set up a lot, including trips lasting up to a week and more and it works very well.
G
