From the description, it sounds like the center piece of the sight may be missing. There should be a small sliding piece held by a very small screw in the rear face of the sight (if it has the factory sight). That slider piece can be used for fine adjustments to fine tune the elevator steps.
The factory sight should look like this one.
http://www.gunauction.com/search/dis...temnum=4857661
Most simply regard the elevator steps as a way to sight the gun in and just move to whatever step is convenient and call it good. In The Winchester Book, Madis mentioned that Winchester had different elevators for different cartridges, and the lowest step was supposed to be for 50 yards, with each step adding 50 yards in range out to 250 or so yards, depending on how many steps the elevator had. I procured a couple of the older style elevators that had the long thumb piece, with patent dates from 1904 I believe. I think they give more range of adjustment than the later very small elevators. I sighted one rifle in at 50 yards on the lowest step by filing the front sight to zero, and with the sight set at the highest elevator step, it hits the 300 yard plate holding right on. Winchester had a wide range of front sight heights available for different models (even recently), it makes sense if the sight was intended to be used as Madis described.
I agree with the receiver sight idea, most of mine have them. I like the Lyman No 56's, but they're getting harder to find. That 1960 gun should be drilled and tapped from the factory for a side mounted receiver sight. I like the sourdough front blade to go with them.