If you look at chain tool, it has "teeth" to put the chain link on, one in front of the other. The one nearest the pushing pin is thicker than the farthest one.
When using the tool to re-insert the link pin, you use the farthest tooth to sit the chain and push the pin in. Once you've done that, you get a tight spot as you've said, so move the link to the nearer tooth and with small turns of the pushing pin, you can spread the links and thus ease the tight spot.
If you look at chain tool, it has "teeth" to put the chain link on, one in front of the other. The one nearest the pushing pin is thicker than the farthest one.
When using the tool to re-insert the link pin, you use the farthest tooth to sit the chain and push the pin in. Once you've done that, you get a tight spot as you've said, so move the link to the nearer tooth and with small turns of the pushing pin, you can spread the links and thus ease the tight spot.
Hope this helps