I would tear it appart and look at the peices to see if any are worn, then go from there there is a bushing kit out there just look for it, or upgrade to a new shifter ... there has to be one out there that will fit ... but i would first investigate to see what the problem is
From Ford Service Manual for 62 Fairlane 3-speed, same for 63.
Gear Shift Linkage Adjustment:
If the transmission shifts hard or will not engage, the gear shift rods may need adjustment. Move the selector lever through all shift positions to be certain that the cross-over operation is smooth. If not, adjust the gear shift rods.
1. With the transmition in neutral, loosen the lock nut on each lever.
2. Center the levers in the retainer at the lower end of the stearing column. The shift handle should be approximately 10 degrees above horizontal.
3. Tighten the lever lock nuts.
4. Check shift for smooth operation. Be sure that the shift levers do not strike the ends of the retainer slots before the transmission detents are engaged.