I have a 1996 mercury cougar. The fuel pump is not coming on (intermittently). The radiator fan ( which hadn't worked till the fuel pump problems) comes on and stays on whenever the fuel pump isn't working..no matter if the cars cold or has been sitting the second the fan cuts off (intermittently) the fuel pump starts working again. I unplugged the fan but after a few days the fuel pump problem resumed. .

I suspect that if you do not have electric fan/s then the fan clutch or viscous fan hub has failed
indications are over heating at idle , stop lights , slow traffic, high rpms at slow speed but cools down as soon as there is enough road speed to put air through the radiator.

Fuel pump relay located
constant control relay modul right front of engine compartment on upper radiator suppor

is it losing water? if not is your fan kicking on? if losing water, place piece of cardboard under veh at night to help pinpoint where water is leaking. if not losing water and fan does'nt come on check fan relay. if loses water and don't see leak over night on cardboard, then water pump is leaking thru weep hole but only after heats up and pressure is on need to replace water pump.

1) low water in the overflow canister. always add water/antifreeze there not directly in the radiator. 2) does the fan at the radiator turn or is running when engine is warmed up?
3) if so then check you belts on your engine? are you missing any?
4) could be bad water pump. do you or have you seen small drops of water coming from the engine area other than the radiator? and other than the A/C unit when car is turned off?
5) when checking fan/s you most like have two fans. One for normal and two come on if you are using air conditioner. make sure both fans are working with A/C on. That should do it.

I would think it's a problem with (CCRM) constant control relay module . This module has relay's inside that control the fuel pump , cooling fans , A/C compressor clutch , PCM - engine computer power relay . And thier are electronics involved , solid state circuity . Google CCRM
found this helpful answer from mercury mechanic on JustAnswer com

Remove the fuel tank

Check your local library for repair manuals...

inside gas tank

Further check out is needed but sounds like the starter is about shot.

My 1999 Cougar started running hot and I took it to my mechanic and he found that the water pump belt was broke. Replaced belt and all is well. Hope this help.