If the fuse is good, that means power should be getting to the relay. If the relay is good, it will close the circuit when the key is on and open the circuit when it is closed. When you are 'probing' around you are making the circuit think the key is on.
I think you might be chasing the wrong thing. You really need to get a multimeter in there to check things out. If your fuse is good then there should be 12v (or whatever the fuel pump draws) all the way to the relay. You also need to know what you're jumping. A relay has two different circuits running through it. It is simply just an electronic switch that controls a high voltage circuit with a low voltage signal. One side will have the main load (12v) and the other side, which I like to use a technical term and refer to it as the 'control'

(much smaller voltage which runs through the coil). Since the only purpose of the relay is to tell the pump to turn on or off, it is going to be a SPST (single pull single throw) relay (switch). When the key is turned on, it is supposed to send a signal to the control side, flipping the switch (closing the main load circuit). The relay is either going to fail open, or closed.
Assuming your fuse is good, then that means you have 12v going to the relay, which also means that you jumping the fuse on that circuit makes no difference. That means that your probing is happening on the 'control' side of the relay, which puts the problem on that same circuit. Since you said you jumped a wire before the relay causing the pump to work, that means your relay is still good.
Provided all the info you gave is correct and that I interpret it right, the problem is not the relay, but instead an open somewhere in the circuit from the key to the relay. Take note one where you jump from/to and narrow the area down.