"Tango Bravo" has is correct but let's expand on the troubleshooting steps.
1) Assure you have not overfilled the unit. (*Total joking
)
1a) Open the trunk and inspect the Brown (Tan) wire that is connected via in line connector near the latch support. Where it penetrates the trunk floor. For grins disconnect and clean (Isopropyl Alcohol or Deoxit 5) & inspect connections both the Packard Series 56 terminals and the plastic connector male and female plugs. and reconnect using a small blade screw driver. (Recheck gauge operations.
& /or
1b) While this connection is disconnected get a (VOM) Volt Ohm Meter digital or analog and set to resistance and test the wire that goes down thru the trunk floor to the sender ie. (tan- brown) thru the variable resistor to ground. Expected range in Ohms is 0-90ohms. 100 ohms to Infinity will cause dash gauge to indicate way full indication thus it overdrives dash gauge past full. Zero ohms should drive the gauge to Empty & 45 ohms appx 1/2 etc.
If no continuity to ground is found;
2) Jack up car secure etc & Remove Right (Pass) Rear tire.
3) Inspect the grounding connection of the circuit the black ground wire from the sender soldered or crimped to the body of the sender-pickup assembly. The lug is terminated at the right side gas tank strap forward hinge point. Remove the sheet metal screw, clean and polish the lug and the body for good ground surface interface, Inspect the wire and lug barrel for broken strand or corrosion etc.
If no ground found after verifying termination;
4) Try to determine the 90 degree molded connection is secure on the top pin of the sender. This is Very difficult to do with the rear end, muffler & tank in place. This may require "DROPPING" the tank.
These two areas are the predominate areas of concern;
Next up would be the trunk wiring extension inline connection in the driver side qtr and issues under the dash related to the connection or further inline connections to the gauge or the 12VDC hot supply (Pink) Additionally the gage ground as TB stated.
Start aft (easy areas then work your way forward.)
I have a pegged gauge on my 68 right now (* I have been playing Fuel Tank Lotto for a while and got stranded once) & I have narrowed it down to the Back of the car by using a Fuel Tank Simulator I made.
JIM