Here's a nice picture of your movement. Either a gear is missing some teeth or the second hand gear is raising up and not engaging properly. Either way, it's an easy fix when you have it serviced. The small center gear is for the second hand. When you lay if flat face up does it move correctly?
I wouldn't doubt if one of the teeth, or a few, broke off of that gear, as the watch hasn't run in 20 to 30 years. I took it to my watch smith this morning and he looked over the movement for me, said it looks OK, but there are signs of rust on the screws that hold the movement in place, which I knew about, but didn't think anything of it. He said other than that, a servicing should fix the problem with the second hand.
It's weird, the whole second hand thing. I put the watch on my dresser last night (usually sleep with my watches on, weird, I know), and the second had was at 45, and when I woke up, it was around 35. So the second hand does advance, just not as it should. It does the same thing when it's on my wrist too.
But it doesn't move normally when laying down flat.