I profess no knowledge re. Professor Longhair...

I would recommend "Live On the Queen Mary".

I like his first four albums.
If you like these also have a listen to Doc.John.
Regards Rob.

I'll second Onhwy61: Queen Mary is one of my all time favorite recordings.
FWIW Prof spawned a generation (or two) of blues piano players in N'awlins. You may want to check out James Boooker's "Lost Paramount Tapes" which is my favorite record from any of Prof's "disciples".

Check out the Aligator Records compliations as well.

Disregard everything listed above. It makes no sense to get into latter 'Fess" first.

There are three early material compilation albums that one should seek out:

"Professor Longhair New Orleans Piano" on Atlanic label.

"1949 Professor Longhair" on the Classics (French) label.

"Mardi Gras in New Orleans" on the Proper (British) label

Me, I'd go with the Proper disc. Twenty-three tracks with good notes and photos for under ten bucks!