JMHO; It sounds like the engine ECM is reverting to cold start ignition timing/ fuel injector mixture program , on every start up; rather than reading the current engine temp. and running the correct program.
The cold start program , replaces the automatic choke operation of an engine with a carburetor.
Have you tried , idling the engine for about two mins. after a hot start , before trying to drive, to see if the noise isn't apparent ?
99DSDP 3884, Freightliner, XC, CAT 3126B, 300 HP /ALLISON 3060
2000 Caravan toad, Remco & Blue Ox.