During the cocking cycle the slide release engages a stud on the hammer if the trigger is held back. That is on the "slam fire" types. Ithaca just removed the stud on the hammer to prevent the "slam fire" feature.
You would have to add a small stud on the right side of the hammr so the slide release can hold the hammer back till the action closes. Watch the slide release when closing the action, it pop's into place locking the action closed. If you take the gun down you will see a small hook on the slide release, that small hook is the sear for the fire on closing (slam fire).
I have two M37 DSPS Ithacas. According to what I have read, one is a mere 12 months older than the other. (One is ~1983, the other ~1982, IIRC.) The older one slam-fires but the newer one does *not*. The things is, I seem to remember seeing the "stud" to which you allude on *both* guns. (I haven't actually had both guns disassembled at the same time.)
I had thought that the slam-fire facility had been disabled about that time and that I had two guns that straddled that era. Now, though, I reckon that *one* of 'em is malfunctioning...