OEM mechanical pumps may not be the best but they can pump a lot of gas in a short time. Three revolutions of the engine should fill the fuel filter and the carb float bowls. It sounds like you have a blockage in the fuel line but the fuel pump could also be bad.
Remove the body to fuel pump line. When lowered below the tank height, fuel should free flow (gravity feed) from the line. Plug the line temporarily after doing the check. If gas flows freely from the body line, then replace the fuel pump. If it does not flow freely from the body line, then the fuel sender screen in the tank is the next thing to check. The screen in the tank can become clogged with old junk (corrosion) in the tank.
Early Mustangs (through mid '65) were originally fitted with a Carter fuel pump having integral filter. The filter screws onto the bottom of the fuel pump. These pumps are still available and can retrofit to any 289/302/351w. I like them because no other fuel filter is required and it makes for a compact fuel system.
In late '65 Ford started installing Carter fuel pumps with separate in-line fuel filter. This is commonly found on '66 and later V8 cars. The fuel pump is dimensionally the same but does not have the screw on filter at the bottom of the pump.
Either of the above are appropriate to your '66. You need both a pump and a fuel filter and so either satisfies the need. Here's a picture of the two types:
http://johnsmustang.com/Catalog/234/