Yes, I had a long reply prepared and then my computer crashed! Anyway, Jack is someone I've been listening to a lot over the last decade and I reallyl see his work as a whole, and there aren't many "just start here" moments.
The album(s) with Hackett ARE a good place to start and they are on cd from Collectables, and a good recommendation! I also really like "Misery and the Blues" which was finally reissued on Verve in the LP By Request series. It's darker than some of his efforts and yet I really consider it one of his best "official" albums. Also worth sampling are the releases on the Vernon Records label; live mateiral from various periods and great notes and engineering. "Meet Me Where they Play the Blues" is also a nice cd from OJC, and another good cd to pick up is the Bethlehem cd. . . .
Jack was not always consistently great, and there are some recordings that have dull moments. I've actually learned to like the big band recordings but it does take time. And I love a lot of sideman appearances in the early thirties. . . . I like his work with Goodman and with Ben Pollack and you can find a lot of these on cds, but it would depend on your tolernace level/like of hot dance and thirties pop music. . . ! If you do like this sort of music, the Classics may be a good route to explore those released under his own name.
Tea is one of my very favorite artists. I'm glad you have become interested in him! There are some similar threads on the board too you may wish to seek out.