On a '64 you will be able to remove the reverse detent plunger and check it and its spring without pulling the engine/transmission. Look below your oil filter and you'll see a hex bolt on the gearbox casing. Loosen and remove it. Under the bolt wil be a spring and plunger. Check the condition of each. If the spring is broken or the detent looks chewed up, order a new set of those parts.
Beyond that you get into transmission specifics. Is this a magic-wand or remote-shift box? You said it's all tight at the gearlever end. The fact that this is becoming a problem between 3-4 may indicate a problem with the linkages on the backside of the gearbox. These (and the condition inside the box) are hard if not impossible to check with the engine in place.