Jeep fuel system

I have 1995 Jeep Wrangler with a 2.5L 4 cylinder. When cold Jeeps fuel pump will not come on to pressurize system. If you sit and crank the engine until the battery is almost dead the relays start to click. When the relays stop clicking, you can hear the fuel pump kick in and pressurize the system. Wait about 60 seconds and it will start right up. Fuel pump and filter are new. Battery new. When it starts it runs perfect, smooth idle, no hesitation. Had alternator checked, 4 Passes in a row. Have run out of ideas.