International 300 utility

Hi, new to this so please forgive if I goof up. I have a 56 300 utility and have always seem to struggle with starting it. Still a 6 volt, new battery, new regulator, rebuilt starter & generator by rep. shop, new push button switch and key switch. Tractor is still REALLY slow turning over and sometimes smokes from area around push button . I was looking at replacing solenoid. Any help would be greatly appreciated.

sounds (looks like) high resistance at push button, getting it hot and smoking. Clean cables with wire brush or steel wool at push button connections, and make sure they are tight. If they still smoke and get hot, cable ends may be corroded under connector, replace or clean cable crimp connector.

The push button is a low current device, less than 4 amps. It is used to energize the starter relay. (That little film can sized device on the side of the starter) if it is smoking at that push
button, there are two possibilities. One is that the button is going bad, or has a bad connection on one of its terminals. Or two the starter relay is going bad and causing heavier current in the button than intended.
Easy checks to make:
Using a piece of 10ga copper wire about a foot long, (with the tractor in Neutral and brakes on!!!) jump between the relay post connected to the battery, and the small terminal on the relay near it. It will spark a little, but if the relay is good the tractor engine will begin to turn over. (don't connect the 2 big terminals together with that small wire! it will get way hot and melt!)
If the tractor spins well the push button needs to be either replaced new switches can be poor quality, or the terminals on it need to be cleaned/tightened and shrink taped to seal out moisture.
If the tractor spins, but slowly, the cables might be incorrect size. They need to be 0gauge wires. These wires are about 3/8" of fine stranded copper with soldered on terminals. Typically prior owners use automotive cables of 4 gauge or so that are way too small. 6v systems use twice the amps of a 12v system and must have thick cables.
If the relay seems to not work well, or requires more than one attempt to engage the starter (series of clicks) replace it. Be sure to connect the new one with the same connection locations. Using 10-40 or 10-30 oil in it will also help in cool/cold weather. Jim

Upon further review, there is NOT a selonoid on side of starter , nor can I find one anywhere! Is this my problem-someone wired it wrong? Cable runs from battery to one side of push button , then from other side of button to lug on side of starter, not to a solenoid.

A system can work like that, H and M Farmalls were (some years) wired in that way.
However!!
to do that the push button switch needs to be capable of 400 amps. The wires need to be the 0gauge I mentioned, and the switch will not fit in the dash. A Relay is easy to wire. A 12 gauge wire from the load side of the amp meter goes to the rubber push button near the key. from the push button a 12ga wire goes to the starter relay mounted on the starter, (small terminal)
The Negative Battery cable (non-grounded) attaches to the relay (post closest to the small terminal) the Opposite terminal attaches to the starter stud. Usually this is a strip of copper bent to connect (Buss Bar) for the 2 inches or so. Making one from a piece of 1/2" copper tubing flattened to allow drilling holes to fit the studs has been done to mimic a original non existent link. Jim

Thank You! I have located a switch but couldn't get there before they closed. Does anyone know where I could locate a diagram so I could make sure when I install the switch nothing else has been done that is different from how it should be and I fry something? Thanks Again! Mike

Have you found a switch, or a starting relay (some call a solenoid, though it is a relay) The wiring is easily found in a IT manual, or IH owners or repair manual. Available here on YT. Mine is 700 miles away. Jim

Here is a Basic start. Jim

Thank You for all your help! I can see I have some work ahead of me! My tractor does not have original ignition switch or push button and there are more wires going to my ignition switch, I am also missing the fuse link completely. How important is that? I guess I'll start with researching some parts and pricing as well!

we had a 300U when I was a kid. It was a grouchy cold natured beast. Everything you are describing is normal. The 300s were IHs first move towards a more modern tractor after 15 years of H and M production. Do yourself a favor and convert it to twelve volts, which means putting an AC Delco alternator on it and new work lights. I did this to my SC Case for $100.00 with all new stuff from advance auto. You will be much more happy with your tractor. Our had a loader on it, we would put a block of wood under it when we shut if off so if we had to pull it to start (often) the bucket would not be on the ground.