Nice car. Someone before you hacked that carburetor out of ignorance and/or desperation. It's not a '60 carburetor, it's a thrown-together "remanufactured" item with parts from various '64-'67 carbs, and it's missing its choke pull-off. One of your pics shows a '73-up choke thermostat (down on the manifold), which might or might not be wired up in a workable manner. Other photos show a manual choke cable kindasorta attached to the carb—can't tell which setup is the car's current setup.
Your last pic shows a vacuum hose leading from the driver side rear of the carburetor (where the choke pull-off hose should go, if there was a pull-off present) to somewhere outside of the photo frame. Where is the other end of that hose?
(To answer your question: yes, the '60-'63 carbs had a choke pull-off mechanism, but it was internal to the air horn, in the form of a metal piston with a link to a little bracket on the choke plate itself. The external diaphragm-type pull-off came in 1964)
Good carbs to suit the rotating-rod throttle linkage used on these pre-'67 cars have grown difficult to get hold of. You might be able to get this carburetor into reasonably workable condition. You'll need a choke pull-off, which won't be terribly difficult to get hold of. You'll need a fast-idle cam, a fast-idle screw and spring, a fast-idle link, and a choke pull-off link, which will call for a "Wanted" ad here and on slantsix.org.
I see other notable things in your pics. From the high placement of your alternator, it appears this car has power steering—nice, and not very common.
Look with suspicion on that '72-up alternator, though; it is very likely a higher-output item than the original 30-amp unit, and that's an electrical fire waiting to happen. The car's wiring was—just—adequate for the original alternator, and circuit protection (fuses, etc) is almost nonexistent on these early cars. Let your battery develop a dead cell, or the battery's fine but flat because the lights got left on and you jump-start it, or let there be some other condition that causes a big drain, and things are going to burn. Best upgrade the charging system wiring and add some circuit protection (and start new threads about those things).
The starter solenoid has been relocated; originally it was perched on the starter motor. Perhaps the starter is a non-original item of one kind or another; can't see it in these pics.
Intake and exhaust manifolds are '73-up items.
Wallowing ride: way-past-due shock absorbers and likely flabby springs. Put a pair of these up front and a pair of these in the rear and that should greatly reduce the seasickness, then you can get after checking and addressing the springs and the rest of the suspension and steering system.
Carburetor operation and repair manuals and links to training movies and carb repair/modification threads are posted here for free download. Tune-up parts and technique suggestions are in this post. Also see this thread.
