I am sure its worn and needs work, but you might reduce the symptoms by trying different oils. Honda HP4 reduced my vintage shifting issues significantly, as did preshifting.
I will need to tear it down next winter for sure and make it right.
Problems shifting to 2nd- Shift Linkage adjustment?
You have to order it online or from a cycle store.
I have no idea why its special, nor can I say for sure that it is.
It helped shifting for lots of classic honda riders, and it did for me.
But it wont subsitute for cracking the cases and fixing the problem.
I see it as something more effective than putting addidtives on shot rings, but not a real fix.
Something to try until you can do a tear down.
I am running 10/40
I have no idea why its special, nor can I say for sure that it is.
It helped shifting for lots of classic honda riders, and it did for me.
But it wont subsitute for cracking the cases and fixing the problem.
I see it as something more effective than putting addidtives on shot rings, but not a real fix.
Something to try until you can do a tear down.
I am running 10/40
I don't think that the problem is the oil. What I think it might be is the shift drum. There are slots machined in the drum to move the shift forks as the drum rotates. For what ever reason, the position that corresponds to second gear results in the configuration of the slot to have a peak with a somewhat sharp point on it. Years ago I experienced a problem similar to what you describe and I found that the peak had broken off the shift drum and this prevented the shift fork from completely engaging second gear. Replacing the drum resolved the problem. Unfortunately the cases have to be split to get to where you can diagnose whether this is the problem or not.