What sized engine? "A series" or the 1500? I looked in my Hayes and the 1500 is rated at 800 rpm at idle, the 1275 is 1000 rpm.
I don't know for sure about the 1500 carb, someone else will have to chime in if thats what you have, but the older A series carbs wear the throttle shaft and carb body, creating an intake leak in the side of the carb. With no pilot jet in the design, this won't draw fuel very well at idle, and those motors are happier at about 1000 -1200 rpm. A replacement throttle shaft cleared up the wear for me, the bodies were not that bad.
Other issues that can make a car hard to tune at a low idle are---
Carbs way out of sync.
A slide that will not descend properly (sticks)
Float level set wrong (low)
intake leak(s) from - gaskets, punctured distributor advance or emission device diaphrams, and worn throttle bodys/shafts.
a badly worn needle and or needle jet. (if you can feel a ridge on the needle, it needs replacing) (worth checking but unlikely)
Carb overheating causing the fuel mixture to change (typically noticed on a hot day in traffic, but not usually as severe as you describe)
Very tight valves (unlikely, you have to have about zero clearance to have an idle issue)
A very tight motor due to improper machining tolerances (A new motor only and the starter will not want to turn it easily) (even less likely)
Unless the air filter is totally clogged, another filter won't make any difference at idle and neither will a new exhaust system. These only come into play under higher rates of flow. You could possibly need a different needle, but the stock needle should be close enough and you should be able to adjust it for idle.
you can check for intake leaks at the carb and elsewhere with an aerosol can of brake/carb cleaner or WD-40. The WD- 40 will make the idle go up and smooth out as it plugs the leak, but is messy. The brake/carb cleaner will usually make the idle worse as its sucked into the carb and interferes with combustion, but it cleans itself up.
On a side note: !!!NEVER use any chlorinated solvent or one that may contain trichlorethelene to do this!!! Under the right conditions (heat and pressure) it can ignite aluminum, albiet slowly and release chlorine gas as well as damage the aluminum, in this case, the piston crowns. I once saw a mechanic spray some down the mouth of a carb on a motor that was already hot. The gas ran us out of the shop for about 5 minutes. I have personnaly seen the stuff smoke aluminum when used as a chip clearing solvent when tapping a hole.
Hope this helps
-Michael