Ah... the ghost of Joe Lucas strikes again.
When I was a young lad I had a Fiat 128 Coupe. It was the prettiest car I ever saw until thenbut anyway.... it had a similar problem.
When the fuel level dropped to a particular point, about 1/3 of a tank, the gauge would "die" and the needle would drop down to read "empty".
If I put more fuel in, to fill the tank above this particular point, the gauge would show the correct level and work normally, until the level dropped to 1/3 again.
What that turned out to be was the gauge sender in the fuel tank. It had a little float on one end, like the inside of a toilet cistern, and at the other end it had an electrical brush contact that moved over a flat wound coil of wire, rather like a piano keyboard (is the best way to describe it).
What happens is, as the float goes up and down at one end, the brush runs up and down the flat wound wire at the other. The position of the brush on the wire creates a higher or lower resistance and that's what the fuel gauge uses to work out how much gas there is.
Anyway, the flat wound wire had snapped at one point (the brush eventually wears it out) and when the gauge reached that point, the contact was broken and the gauge needle collapsed.
That doesn't sound like your problem but I just thought I'd mention it,