The mechanical pump could be getting weak with a leaking check valve. I've had a couple over the years that would run fine down the road but would lose prime when sitting for a few weeks.

When installing an electric fuel pump be sure to:

Install an oil pressure switch to shut off power when the engine stops. The switch needs to have at least 2 terminals and conducts when the oil pressure is up. ( Some have a 3rd terminal for the oil pressure light ) GM used these on late 70's / early 80's carburetor cars for the electric choke function. Have a look at a 78 to 82 ish Buick Regal with a V6 for an example.

Installing an inertia switch is a good idea too in case the car crashes, breaks a fuel line and engine continues to run. This switch can be found on fuel injected Ford products from the mid 80's and up to 2000 + or so. They have a round red button or a rectangular white one. Look in the quarter panel trunk area of a car or passenger interior fire wall / kick panel of a pickup.

The down side is you can no longer crash a barricade. On my Ford pickup, with malice aforethought, I've bypassed the inertia switch / anti theft starter cutout / clutch switch to maintain some mobility if things turn dire.

I use the " looks like a fuel filter with 2 wires " type solenoid pump. These pumps make a jiggling noise when pumping especially when priming / flowing fuel so mounting them on soft mounts may be needed. Carter and Airtex make them as do a few others, I'd stay away from an unbranded China Amazon / E bay pump though. Get the " import low pressure " unless the shop manual calls for the USA higher pressure unit.

The pumps should be mounted near the tank as they are better for pushing than pulling. A rear mount reduces vapor locking tendencies as well.

It is possible to use this pump in addition to a regular mechanical pump as both will move fuel if one outruns the other.