I'd like to pitch in and add some info. The pan and (especially) gasket info and part numbers in here are worth knowing even if your transmission is a '64 or later.
The pre-1964 Torqueflites used an external, in-line transmission fluid filter as an optional accessory in '60 and '61, standard equipment in '62-'63, not a flat in-pan filter such as the 1964-up transmissions used. In place of the in-pan flat filter, the early cars had a fine-mesh metal screen. If your car is so equipped, you will find the black, canister-shaped filter clamped to the engine block right near the level of the oil pan rail on the left side, with the trans fluid pipes threaded into each end of the canister. These canister-type filters are no longer manufactured. New Old Stock items can sometimes be found, but are usually quite costly. There are aftermarket universal inline filters, but they look cheesy. Why not eliminate the hassle once and for all? You can easily convert to the '64-up in-pan filter, making future transmission service easier and less costly.
There are a few different in-pan filters that will physically fit, but on a pre-1966 transmission, you need a filter with two fluid ports in it. The '66-up filters have only one, and if you use them on a '65 or earlier transmission, you'll starve the rear pump and damage it; it'll get noisy.
For this project, you'll need:
•A 1964-1965 transmission filter (with two ports). NAPA or ATP 19715 or 17956 trans filter/gasket kit contains the 2-port filter. Other numbers include Wix 58656 and Fram (yuck) FT-1015A.
•A 1964-up transmission pan or a brand-new pan. Nice new deep pans with unwarped rails for the A904 (i.e., the stock A500 pan) can be had from the dealer under p/n 52118 779AD, and they even include a spiffy magnet to catch metallic shavings. Whether you go that route or install a used '64-up pan, save yourself a bunch of current and future hassle: discard the floppy cork or rubber pan gasket that comes with the filter kit. Instead, from the Chrysler dealer get the really nice double-seal, reusable rigid pan gasket P/N 4295 875AC. These part numbers are for A904 transmissions, which have a roughly square-shaped transmission pan with one corner cut off. The larger A727 transmissions take pan p/n 5211 8780AD and gasket 2464 324AB. The doughnut magnet itself is 3681 601.
The reason why you need to change the pan is that when they changed to the in-pan Dacron filter for '64, they embossed a depression in the pan to provide sufficient clearance between the pan and the Dacron filter, which is thicker than the '63 and earlier pickup screen.
If you go with the deeper late pan, you do not need a filter extension or any other adaptation. It will take a little more fluid to bring the dipstick up to the "Full" mark, but that full fill of fluid will be at exactly the same level relative to the filter, no matter the depth of the pan.