62 Valiant Speedometer

The car (and therefore the speedometer) is nearly half a century old. It's an extremely good bet that the speedo bearings are worn out. Taking out the speedo and examining it is a good idea, but before you spend too much time on it, grasp the stirrup and the cup (one at a time) and try to wiggle them side to side. If there's any perceptible side to side movement in either the cup or the stirrup, the speedo needs rebuilt; continuing to use it (fresh oil or no fresh oil) is to consign it to certain death when the bearing wear goes just far enough that the stirrup grabs the cup and SLAMS it hard against the stop.

Careful not to over-lube the speedo cable; it is better to err on the side of too little than too much grease. see here.

Also, the "universal" speedometer cable kit replaces only the inner core; the (usually worn) casing is kept and if it is as worn as it usually is on such an old vehicle, it will quickly fail the new cable core. These pre-'66 cars take a cable with a special transmission end; you can usually get a replacement complete cable (with casing) from Pat "Torqueflite Patty" Blais or from Andy Bernbaum, www.oldmoparts.com .