Ideally What I am looking for is a 1935 to 40 Cadillac, LaSalle, or Packard Convertible sedan, maybe even a Buick. I am not after a #4 or #5 car, way out of my range. While the car may not run, it should need only attention, or if the drive train is missing, that "could" work. The body needs to be complete & solid; I can handle the mechanical stuff. If you have or know of one for sale, call me or email. (972) 754-0344 – kenlobsinger@hotmail.com.
Personally, I would be looking for the best car that I could afford. That way the work is basically done for you. Figure in the cost of parts and availability for a car that old. Maybe you are looking to build a Resto-Mod, or are you looking to build a correct era car? Maybe even a (cring) Rat-Rod.
Geoff will tell you all about doing his '35 Cadillac. I think he worked on that car religiously for about 4-5 years. Now, it's more reliable than any new car you could buy.
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Finding a Pheaton / four door convertible of any make is going to be tough.
He probably doesn't still have the POS but someone name Chuck or Charles P. has/had a 1940 LaSalle convertible sedan.
RUN away from it if offered to you. Supposedly the last one built.
Be careful of any '30s car, the main body structure and doors are wood, the sheet metal is tacked to it. Wood rot is very common in those cars, it is repairable/replaceable but there aren't too many doing new wood body structures and door wood (although they are out there), time consuming, and expensive. My '35 survived much because it was probably garaged and spent from 1965 to 2007 (when I bought it) stored in a tractor-trailer van body.
1934 and 1935 Cadillacs are nearly identical (except for the bumpers) and not many were made back in those depression years.
Easier to find a very similar looking '34/'35 Buick but all are still hard to find and have the wood problem. A convertible sedan (4 doors) of any year '34 to '40 will be very hard to find and difficult/expensive to source certain parts for restoration. Search on e-Bay (that's where I found my sedan), Hemmings, Bring-a-trailer, Facebook's "For Sale" section; Hagerty Insurance also has a cars for sale service.
Some raise their eyebrows and scoff at e-Bay, but I found a nice '60 Cad. coupe (I restored and flipped), my '35 Cadillac (Big Red) that I resto-modded (now 40,000 + miles and never to be sold), my '73 Cadillac Caribou (sold, bought back, and never to be sold again), a 1950 Jaguar MK V sedan that I resto-modded and ultimately sold to our local regional AACA president (I see it often). Keep looking everywhere you can think of, be prepared to lay critical eyes on the car, and be patient.