There are numerous aspects to watch out for when buying a second-hand Morgan. Many people recommend reading "Moggie" by Colin Musgrove, as it contains a wealth of advice.
Be sure to check the state of the inner wing valances at the rear, where they sit against the side of the bulkhead. If they rust through here, there is likely to be a hole through the side of the bulkhead too. If you have to replace the inner wings, the outer wings will probaby also have to be relaced, as matching the profile of the wings to the inner valances will be next to impssible.
Check the state of the door hinges by trying to lift the door whilst it is open. If the hinges are tight, but movement is still detected, then it means the woodwork that the hinges are mounted on is rotten
A key point is a possible kick-up under the back axle. Run a straight edge under the bottom chassis rail at this point and see what you find. If the kick up is too obvious you could have lots and lots of problems, including wood rot or separation at the bottom of the inner rear wheel valances and the chassis wearing through or cracking at around the point where the side rails narrow under the axle.
Also have a look at the cross-members, particularly the rear one. On the Series 1 these are "top hat" section, but from the Plus 4 on they are boxed in and the bottom plate can get quite rusty. (Suggested by John Merton).